5.1 Exponents and Scientific Notation

Transcription

1 5.1 Exponents and Scientific Notation Definition of an exponent a r = Example: Expand and simplify a) 3 4 b) ( 1 / 4 ) 2 c) (0.05) 3 d) (-3) 2 Difference between (-a) r (-a) r = and a r a r = Note: The 1 st says use a as the base, the second says the opposite of the answer of a r. The reason that this is true is because the exponent only applies to the number to which is written to the right of and a is 1 a and therefore the 1 isn't being raised to the "r" power. Example: Name the base, exponent, rewrite using repeated multiplication, and simplify to a single number. a) (-5) 2 b) -5 2 Example: Simplify a) (-3) 2 b) -3 2 c) Definition of zero exponent if a 0 a 0 = Anything to the zero power is 1. Be careful because we still have to be certain what the base is before we rush into the answer! Example: Simplify each by removing the negative exponent. a) 2 0 b) ( 1 / 2 ) 0 c) -2x 0 d) e) 2x 0 y 0 d) (7 + x) 0 Y. Butterworth Ch. 5 Martin-Gay/Greene 1

2 Product Rule for Exponents a r a s = Example: Use Product Rule of Exponents to simplify each of the following. Write the answer in exponential form. a) x 2 x 3 b) (-7a 2 b)(5ab) c) d) y 2 y 3 y 7 y 8 Example: Simplify using the product rule y 2m + 5 y m 2 Definition of a negative exponent (Shorthand for take the reciprocal of the base) a -r = Anytime you have a negative exponent you are just seeing short hand for take the reciprocal. When a negative exponent is used the negative in the exponent reciprocates the base and the numeric portion tells you how many times to use the base as a factor. A negative exponent has nothing to do with the sign of the answer. Example: Simplify each by removing the negative exponent. a) 2 1 b) ( 1 / 2 ) 1 When a negative exponent has a numeric portion that isn't one, start by taking the reciprocal of the base and then doing the exponent (using it as a base the number of times indicated by the exponent). Example: Simplify each by removing the negative exponent. a) 2 2 b) ( 1 / 2 ) 3 Don't let negative exponents in these bother you. Copy the like bases, subtract (numerator minus denominator exponents) the exponents and then deal with any negative exponents. If you end up with a negative exponent it just says that the base isn't where it belongs if it's a whole number take the reciprocal and if it is in the denominator of a fraction taking the reciprocal moves it to the numerator. Let's just practice that for a moment. Example: Simplify each by removing the negative exponent. a) a 2 b) 1 b 3 Y. Butterworth Ch. 5 Martin-Gay/Greene 2

3 The Quotient Rule of Exponents a r = a s Example: Simplify each. Don't leave any negative exponents. a) a 12 b) 2b 5 c) x 8 a 7 b 3 x 2 d) 2b e) (a + b) 15 f) 2a 3 b 2 16 b 3 (a + b) 7 4ab 3 Now, let's practice the quotient rule with negative exponents. You have 2 choices in doing these problems: 1) Use the quotient rule on integers (can be tricky if your integer subtraction is not what it should be) or 2) Remove the negative exponents and use product/quotient rules as needed. Example: Simplify each. Don't leave any negative exponents. a) a 8 b) 2 c) x -8 a 10 b 3 x -2 d) 2 b 3 e) (a + b) 5 f) 2a 3 b -2 4 (a + b) -7 ab 3 Example: Simplify using the quotient rule 15x 7n 5 3x 2n 3 Our next topic is an application of exponents that makes writing very large and very small numbers much easier, especially in application. Think of scientific notation as what you see every day in written expression of large numbers like in a newspaper article. You might see 5,000, 000 written a 5 million. This is the same thing that scientific notation does, but it uses multiplication by factors of ten, which are decimal point movers. Y. Butterworth Ch. 5 Martin-Gay/Greene 3

4 Scientific Notation When we use 10 as a factor 2 times, the product is = 10 x 10 = 100 second power of 10 When we use 10 as a factor 3 times, the product is = 10 x 10 x 10 = 1000 third power of 10. When we use 10 as a factor 4 times, the product is 10, = 10 x 10 x 10 x 10 = 10,000 fourth power of 10. From this, we can see that the number of zeros in each product equals the number of times 10 is used as a factor. The number is called a power of 10. Thus, the number 100,000,000 has eight 0's and must be the eighth power of 10. This is the product we get if 10 is used as a factor eight times! Recall earlier that we learned that when multiplying any number by powers of ten that we move the decimal to the right the same number of times as the number of zeros in the power of ten! Example : 1.45 x 1000 = 1,450 Recall also that we learned that when dividing any number by powers of ten that we move the decimal to the left the same number of times as the number of zeros in the power of ten! Example : = Because we now have a special way to write powers of 10 we can write the above two examples in a special way it is called scientific notation. Example : 1.45 x 10 3 = 1,450 ( since 10 3 = 1000 ) Example: x 10-2 = ( since 10 2 = 100 and [ 10 2 ] -1 = 1 which 100 means divided by 100) Writing a Number in Correct Scientific Notation: Step 1: Write the number so that it is a number 1 but < 10 (decimals can and will be used) Step 2: Multiply this number by 10 x ( x is a whole number ) to tell your reader where the decimal point is really located. The x tells your reader how many zeros you took away! (If the number was 1 or greater, then the x will be positive, telling your reader that you moved the decimal to the right to get back to the original number, otherwise the x will be negative telling the reader to move the decimal left to get back to the original number.) Y. Butterworth Ch. 5 Martin-Gay/Greene 4

5 Example : Change 17,400 to scientific notation. 1) Decimal ) Multiply x 10 Example : Write in scientific notation 1) Decimal ) Multiply x 10 Example : Change each of the following to scientific notation a) 8,450 b) 104,050,001 c) 34 d) e) f) Note: When a number is written correctly in scientific notation, there is only one number to the left of the decimal. Scientific notation is always written as follows: a x 10 x, where a is 1 and <10 and x is an integer. We also need to know how to change a number from scientific notation to standard form. This means that we write the number without exponents. This is very simple, we just use the definition of scientific notation to change it back in other words, multiply the number by the factor of 10 indicated. Since multiplying a number by a factor of 10 simply moves the decimal to the right the number of times indicated by the # of zeros, that s what we do! If the exponent is negative, this indicates division by that factor of 10 so we would move the decimal to the left the number of times indicated by the exponent. Example : Change x 10 5 to standard form 1) Move Decimal to the Right times. 2) Giving us the number Example : Change x 10-3 to standard form. 1) Move Decimal Left times 2) Giving us the number Example: Write each of the following to standard form. a) x 10-3 b) x 10 5 c) x 10-5 d) x 10 6 Y. Butterworth Ch. 5 Martin-Gay/Greene 5

6 5.2 More Work with Exponents and Scientific Notation This section just takes what is left of exponent rules and scientific notation and discusses those topics. We have power rules and their combinations with other rules to discuss about exponents. In scientific notation, we need to discuss how to multiply and divide and use this in applications. Power Rule of Exponents (a r ) s = or or (ab) s = ( a / b ) r = Example: Use Power Rule of Exponents to simplify each of the following. Write the answer in exponential form. a) (a 3 ) 2 b) (10xy 4 ) 2 c) (-7 5 ) 2 d) ( -3a / 5b ) 3 Note: A negatvie number to an even power is positive and a negative number to an odd power is negative. Now that we have discussed all the rules for exponents, all we have left is to put them together. Let's practice with some examples that use the power rule, the product rule and also the quotient rule. Use the power rule 1 st, then the product rule and finally the quotient rule. Deal with the negative exponents last. Example: Use the properties of exponents and definitions to simplify each of the following. Write in exponential form. a) (x 2 y) 2 (xy 2 ) 4 b) (-8) 3 (-8) 5 c) ( 3 / 2 ) [ ] 2 [ ] d) 9x 2 y 5 e) (2xy 2 ) 3 f) -5x 2 y 3 15xy 8 4x 2 y 4x -1 2 [ ] -2 [ ] g) 9x 2 y 5 h) -5x 2 y 3 i) (2xy 3 ) -2 15xy 8 4x -1 xy 2-3 Y. Butterworth Ch. 5 Martin-Gay/Greene 6

7 Your Turn Example: Simplify each. Don't leave any negative exponents. a) (ab) 8 b) (-2xy) 3 c) x -8 a 10 3x 3 y x -2 [ ] -3 d) (5xy 2 ) 2 (4x -1 y -3 ) -3 e) x 3 x 5 x 7 5x 3 (2 x 3 y -5 ) -2 f) 2a 3 b -2 ab 3 Now, for the next step with scientific notation it can used to multiply and divide large/small numbers. This is really quite easy. Here is some explanation and how we can do it! What happens if we wish to do the following problem, 7 x 10 2 x 10 3 = (7 x 10 2 )(1 x 10 3 ) We can think of 10 2 and 10 3 as "decimal point movers." The 10 2 moves the decimal two places to the right and then the 10 3 moves the decimal three more places to the right. When we are finished we have moved the decimal five places to the right. This is an application of the product rule. Steps for Multiplying with Scientific Notation: Step 1: Multiply the whole numbers Step 2: Add the exponents of the "decimal point movers", the factors of 10. Step 3: Rewrite in scientific notation where the number multiplied by the factor of 10 is 1 but < 10. Before we begin practicing this concept, I want to practice a skill. I want to learn to write a number in correct scientific notation. Y. Butterworth Ch. 5 Martin-Gay/Greene 7

8 Steps for Writing in correct scientific notation Step 1: Write the number in correct scientific notation Step 2: Add the exponent of the new number s factor of 10 and the one at the start. Example: Write in correct scientific notation. a) 14.4 x 10 5 b) x 10-3 c) x d) x 10-4 Example : Multiply and write the final answer in correct scientific notation. a) (3 x 10 2 ) ( 2 x 10 4 ) b) (2 x 10-2 ) (3 x 10 6 ) c) (1.2 x 10-3 ) (12 x 10 5 ) d) (9 x 10 7 ) (8 x 10-3 ) Note: In part c) & d) once you multiply the numbers you have a number that is greater than 10 so it must be rewritten into correct scientific notation by thinking about the number that 14.4 x actually represents and changing that to scientific notation. Steps for Dividing with Scientific Notation: Step 1: Divide the whole numbers Step 2: Subtract the exponents of the "decimal point movers" (numerator minus Denominator exponents) Step 3: Rewrite in scientific notation where the number multiplied by the factor of 10 is 1 but < 10. Example: Divide using scientific notation. Be sure final answer is in correct sci. note. a) ( 9 x 10 5 ) b) ( 2.5 x 10 7 ) c) ( 2 x 10-2 ) ( 3 x 10 2 ) ( 2.5 x 10 5 ) ( 1.5 x 10 5 ) Example: Perform the indicated operation using scientific notation. a) 3000 x b) x 11, x Y. Butterworth Ch. 5 Martin-Gay/Greene 8

9 5.3 Polynomials and Polynomial Functions First, we need to have some definitions. You should already know several, and several may be new, make an effort to learn vocabulary as math is like a foreign language! Constant A number Numeric Coefficient A number multiplied by a variable Term A part of a sum (constant, variable, variable times a constant) separated by addition/subtraction symbol Like Terms Terms which have the same variable component (must be to same powers) Polynomial A term or the sum of two or more terms of the form ax n (an algebraic expression in only one variable) Monomial A polynomial with a single term. Binomial A two termed polynomial. Trinomial A three termed polynomial. Degree of a Term The sum of the powers of the variables in a term. The degree of a constant is zero. Degree of a Polynomial The degree of the highest degreed term within the polynomial. Example: Each of the following form algebraic expressions (some polynomials). I want to discuss each in terms of the above vocabulary: constant, variable, numeric coefficient, and type of polynomial. a) 3x / 2 b) 2x 2 + 2x c) 2(x 5) + 3x d) x 2 + 5x + 5 Example: Are the following like terms? Why or why not? a) -5x & 25x b) 3 / 4 xy & 3 / 8 x 2 y c) -5 & 5x d) 54x / 3 & -54xy / 3 Example: What is the degree of the term? a) b 2 b) 2 b 3 c) x 2 y 2 z d) 1 Example: What is the degree of the polynomial? a) a 2 + 3a + 5 b) 3a + 4 a 3 2a 2 6 c) ab 2 + 3a 2 b 3 + 2a 2 4 Y. Butterworth Ch. 5 Martin-Gay/Greene 9

10 Method 1: Adding/Subtracting Polynomials (Horizontal Combining Like Term) Step 1: Remove grouping symbols (This is really distribute the subtraction!!) Step 2: Group Like Terms Step 3: Combine Like Terms Example: Add/Subtract the following polynomials/algebraic expressions. a) ½ x 2 + 5y 1 / 3 x 2 + 3y b) 2(x 5) + 3x c) [(8xy 2 + 2x) + (-7xy + 3)] (xy 2 + 3x) d) x 2 + 5x + 5 e) (7xy 2 2x + 3) (5x 2 y 4x 9) f) ½ (x y) ¼ (x + 3) ½ y Note: You can not clear an expression of fractions as you can an equation. You must deal with them as they are. There is another way to think about adding and subtracting polynomials. This is columnar addition and subtraction. We must really focus on ordering the polynomial to do this. Ordering a polynomial means putting the terms in order of descending degree. Example: Order the following polynomials/algebraic expressions a) 7x 2 + 9x 3 x b) xy 9 + xy 2 Method 2: Adding/Subtracting Polynomials (Algebraic Expressions) in Columns Step 1: Order polynomials being added or subtracted (leave blanks for missing degrees) Step 2: Remove subtraction Step 3: Stack in columns (like terms over one another, leave blanks where there are no like terms) Step 4: Add Y. Butterworth Ch. 5 Martin-Gay/Greene 10

11 Example: Add/Subtract the following polynomials/algebraic expressions a) (7x 2x 2 + 3) (5 + x 2 2x) b) (9x 2 9) + (x 2 + x + 7) c) ( 2 / 3 x 2 9) ( 1 / 5 x x 1 / 3 ) d) Subtract 3x + 3 from 5x 2 + 4x 7 A polynomial function is a function that contains a polynomial. P(x) = ax n + ax n 1 + ax n ax n (n 1) + a Finding the Value of a Polynomial Function(Evaluating an polynomial/alg. exp.) Step 1: Leave blanks for variables Step 2: Fill the blanks with the value of the variable, the number inside the P(?) Step 3: Simplify the resulting numeric expression using order of operations Example: For P(x) = (x 2 3x + 3) find P(-1) Example: For P(x) = (x 2 5) 2 find P(4) Example: For P(x) = 2(x + 2) 3 find P(x + h) Y. Butterworth Ch. 5 Martin-Gay/Greene 11

12 We can even use this in terms of application problems: Example: The average cost of a college education per year for a full-time student in two year public colleges can be approximated by the polynomial function D(x) = 6.4x x for the years 1984 through Predict how much it will cost in 2010: D(26). Round to the nearest dollar. (Martin-Gay, Beginning Algebra, 5 th Edition, p. 323 #111) Oh, and life would not be complete without the use of function notation in adding and subtracting polynomials. Just substitute and simplify! Example: If P(x) = 4x 2 6x + 3 & Q(x) = 5x 2 7 find the following a) 2[P(x)] + 7[Q(x)] b) Q(x) P(x) Also in review for us this section is the Quadratic and Cubic functions that we saw in 2.1. The reason they are in review in this section is for the interest in their degree. A quadratic function, P(x) = ax 2 + bx + c, is of degree 2 and all polynomial functions of degree 2 will be quadratic and will have a graph that looks like a parabola. A cubic function, P(x) = ax 3 + bx 2 + cx + d, is of degree 3 and all polynomial functions of degree 3 will be cubic and will have a graph that looks like a lazy s. Also under review is the bit of information that we learned in 2.3, about the y-intercept of functions being the constant value. You will need to review these two concepts from these two sections, to complete problems like the following and problems like those found in exercises 89-92: Example: Give the coordinates of the y-intercept of the graph of each equation and describe the shape of the graph. a) f(x) = x 3 6x + 4 b) g(x) = -2x x 3 c) h(x) = 5x 2 2x + 1 d) m(x) = -x 3 3x 2 + 2x 5 Y. Butterworth Ch. 5 Martin-Gay/Greene 12

13 F O I L }These add & are like terms if the binomials are alike. 5.4 Multiplying Polynomials Multiplying a monomial by a monomial is really an application of the properties of exponents combined with the associative property. I will review slightly, but this is not something that should be difficult. Monomial x Monomial Multiply the coefficients Add the exponents of like bases Example: Find each product. a) (5a 2 )(-6ab 2 ) b) (-r 2 s 2 )(3ars) Multiplying polynomials is an application of the distributive property. This is also called expanding. Distributive Property Review a ( b + c ) = ab + ac Monomial x Polynomial Example: Multiply. a) 2x (x 2 + 2x + 3) b) -4x 2 y(x 2 y 2xy + 3y) Binomial x Binomial Now we'll extend the distributive property further and to help us remember how we will have an acronym called the FOIL method. F = First O = Outside I = Inside L = Last ( a + b ) ( d + c ) = ad + ac + bd + bc Example: Multiply. Think about only sum & product of the constants to get the middle & last terms of your polynomial. a) (x + 2) (x + 3) b) (x 5)(x + 9) c) (x 4)(x 3) d) (x + 7)(x 9) Y. Butterworth Ch. 5 Martin-Gay/Greene 13

14 Note: A binomial of the form (x + c) multiplied by a binomial of the same form will always yield a trinomial. The first terms yield the highest degree term(of degree 2), the inside and outside add to give the middle term and the last will yield the constant. Example: Multiply. Remember that it is the outside & inside products that will add & the last & first terms multiply to give first and last terms of the trinomial. a) (x 5) (2x + 3) b) (3x + 4)(x + 5) c) (2x 5)(x 4) d) (2x 5)(x + 3) e) (2x + 3)(5x + 4) f) (2x 4)(3x + 5) g) (2x + 4)(3x 5) h) (3x + 5) (4x + 1) Example: Multiply. (x 2 + y) (x y) Note: This didn t yield a trinomial because the binomials are not of the same form. Example: Multiply. a) (5x + 1)(3x + 1) b) (4x 1)(3x 1) c) (5x 1)(6x + 1) d) (8x + 1)(9x 1) Y. Butterworth Ch. 5 Martin-Gay/Greene 14

15 Note: These are just like multiplying when the numeric coefficient of the x is one. The only differences are that the leading coefficient is the product, not the constants, and you must watch the sign on the middle term a little more carefully. Product of x coefficients gives leading coefficient & sum/difference of the x s coefficients gives the middle term (just watch if positive or neg. since that comes from outside & inside products). Example: Find the product. a) 5x(x + 2)(x 2) b) -3r(5r + 1)(r + 3) Note: The 1 st example is possibly more easily done by doing the product of the binomials 1 st and then multiplying by the monomial. Always remember that multiplication is commutative and associative so you can choose which product you find 1 st. Special Products Squaring a Binomial (Returns a Perfect Square Trinomial) (a + b) 2 = a 2 + 2ab + b 2 or (a b) 2 = a 2 2ab + b 2 Example: Expand. a) (2x + 3) 2 b) (2x 3) 2 c) (2x + 3y) 2 d) [(3x + 2) + y] 2 Product of Conjugates (Returns the Difference of Two Perfect Squares) (a + b)(a b) = a 2 b 2 Example: Expand. a) (3x + 5)(3x 5) b) (2x y)(2x + y) c) [(x + 1) + 3][(x + 1) 3] d) ( 4 / 3 x + 5)( 4 / 3 x 5) Polynomial x Polynomial Multiplying two polynomials is an extension of the distributive property. The first term of the first polynomial gets multiplied by each term of the second, and then the next term of the first gets multiplied by each term of the second and so on and the end result is the sum of all the products. Y. Butterworth Ch. 5 Martin-Gay/Greene 15

16 Of course there is a second way to do the distributive property and it takes care of keeping all the degrees of variables in order. This method looks like long multiplication. All polynomials must be ordered for this method to be successful. I will do one of the examples below using both methods, and all others using just the new method. Example: Multiply using long multiplication. a) (2x + 3) (x 2 + 4x + 5) b) (x 2 + 2x 7) (x 2 2x + 1) c) (2x 2 + 4x 8)( 1 / 2 x + 3) In this last example we have to simplify by multiplying the first two polynomials out and then multiplying that result by the third polynomial. We ll only do this example one way, but I will use one method on the first 2 and then the other method on the result times the last. It is interesting to point out that sometimes it is useful to use the commutative property to rearrange the order thus deciding the two you will multiply first, as some binomial products are much simpler than others! Example: Simplify a) (x + 1)(2x 3)(x + 1) b) (2x + 1)(3x + 2)(2x 1) Y. Butterworth Ch. 5 Martin-Gay/Greene 16

17 x 7 Now for some application of our polynomial functions. 2x x Example: For P(x) = x and Q(x) = x 2 2 find a) [P(x)] 2 b) P(x) Q(x) Example: For the following figure compute the area of the shaded region. Y. Butterworth Ch. 5 Martin-Gay/Greene 17

18 5.5 The Greatest Common Factor and Factoring by Grouping Recall that the greatest common factor(gcf) is the largest number that two or more numbers are divisible by. A GCF isn t a big number at all, it is a little piece of a number and it is the biggest, little piece that all the numbers have in common. The largest a GCF can ever be is the smallest number. Finding Numeric GCF Step 1: Factor the numbers. a) Write the factors in pairs so that you get all of them starting with 1? = # Step 2: Find the largest that both have in common. Example: Find the GCF of the following. a) 18 & 36 b) 12, 10 & 24 We are going to be extending this idea with algebraic terms. The steps are: 1) Find the numeric GCF (negatives aren t part of GCF) 2) Pick the variable(s) raised to the smallest power that all have in common (this btw also applies to the prime factors of the numbers) 3) Multiply number and variable and you get the GCF Example: For each of the following find the GCF a) x 2, x 5, x b) x 2 y, x 3 y 2, x 2 yz c) 8 x 3, 10x 2, -16x 2 d) 15 x 2 y 3, 20 x 5 y 2 z 2, -10 x 3 y 2 Now we'll use this concept to factor a polynomial. Factor in this sense means change from an addition problem to a multiplication problem! This is the opposite of what we did in multiplying a monomial by a polynomial. Factoring by GCF Step 1: Find the GCF of all terms Step 2: Rewrite as GCF times the sum of the quotients of the original terms divided by the GCF. Be sure that your polynomial is the same length as the polynomial that you started with. Y. Butterworth Ch. 5 Martin-Gay/Greene 18

19 Let's start by practicing the second step of this process. Dividing a polynomial that is being factored by its GCF. Example: For each of the following, factor using the GCF a) 8 x 3 4x x b) 27 a 2 b + 3ab 9ab 2 c) 18a 9b d) 108x 2 y 2 12x 2 y+ 36xy xy e) 1 / 7 x / 7 x f) -x 2 + x 1 Note: When the GCF involves fractions with like denominators the GCF of the numerators is what matters and the denominator tags along. Remember when dividing by a fraction it is multiplying by the reciprocal. Note2: It is considered unpalatable for the leading coefficient of a polynomial to be negative, and it is therefore desirable to factor a negative one from such a polynomial. Sometimes a greatest common factor can itself be a polynomial. These problems help it make the transition to factoring by grouping a lot easier. Example: In the following problems locate the 2 terms (the addition that is not in parentheses separate terms), and then notice the binomial GCF, and factor it out just as you did in the previous problems a) t 2 (t + 2) + 5(t + 2) b) 5(a + b) + 25a(a + b) Our next method of factoring will be Factoring by Grouping. In this method you rewrite the polynomial so that terms with similar variable(s) are grouped together. This type of factoring will take some practice, because the idea is to get a polynomial which will have a binomial in each term that we will then be able to factor out as in the last two examples. Factoring By Grouping Step 1: Group similar terms and factor out a GCF from each grouping (keep in mind the aim is to get a binomial that is the same out of each grouping(term) look for a GCF) Step 2: Factor out the like binomial and write as a product (product of 2 binomials) Hint: Trinomials are prime for factoring with the GCF and a polynomial with 4 terms is prime for this method Y. Butterworth Ch. 5 Martin-Gay/Greene 19

20 Example: In the following problems factor out a GCF from binomials in such a way that you achieve a binomial in each of the resulting terms that can be factored out. a) 8x y xy 2 b) 2zx + 2zy x y Note: In b), to get the binomial term to be the same you must factor out a negative one. This is the case in many instances. The way that you can tell if this is the case, look at your binomials if they are exact opposites then you can factor out a negative one and make them the same. c) b 2 + 2a + ab + 2b d) xy 2 + 2y x Note for c): Terms must be rearranged to factor a GCF from a binomial. There are several different possibilities, so don't let it worry you if you would have chosen a different arrangement. [b 2 + 2b + 2a + ab is one and b 2 + ab + 2a + 2b is the other] Note for d): Terms must be rearranged to factor a GCF from a binomial. There are several different possibilities, so don't let it worry you if you would have chosen a different arrangement. [xy x + 2y 2 is one and xy + 2y x 2 is the other] Note 2 for d): In addition to rearranging the grouping the order of the terms can also be rearranged in each grouping resulting in the necessity to see that terms are commutative, when they are added to one another. [xy x 2 + 2y, results in x(y 1) + 2(-1 + y) or if you factored out a -1, then it looked really different x(y 1) 2(1 y), but you can recognize that (1 y) is the opposite of (y 1) and get the situation turned in your favor!] e) 6x 2 y + 15x 2 6xy 15x f) 5x 2 y + 10xy 15xy 6y Note for e)&f): These one is a bit trickier still! It has a GCF 1 st and then factoring by grouping. Y. Butterworth Ch. 5 Martin-Gay/Greene 20

21 Your Turn 1. Find the GCF of the following a) 28, 70, 56 b) x 2, x, x 3 c) 3x 3, 15x 2, 27x 4 d) x 2 y 3, 3xy 2, 2x e) 12xy 2, 20x 2 y 3, 24x 3 y 2 2. Factor the following by factoring the GCF a) 28x 3 56x x b) 9x 3 y 2 12xy 9 c) 15a 2 60b 2 d) a 2 (a + 1) b 2 (a + 1) 3. Factor the following by grouping a) xy 2 + y 3 x y b) 5x 2 + x y 5xy c) 12y 3 + 9y y + 12 Y. Butterworth Ch. 5 Martin-Gay/Greene 21

22 5.6 Factoring Trinomials It is important to point out a pattern that we see in the factors of a trinomial such as this: (x + 2)(x + 1) = x 2 + 3x + 2 x x Product of 1 st 's Sum of 2 nd 's Product of 2 nd 's Because this pattern exists we will use it to factor trinomials of this form. Factoring Trinomials of the Form x 2 + bx + c Step 1: Start by looking at the constant term. Think of all it's possible factors Step 2: Find two factors that add/subtract to give middle term's coefficient a) They will ADD if the constant s sign is positive b) They will SUBTRACT if the constant s sign is negative Step 3: Write as (x ± 1 st factor)(x ± 2 nd factor) ;where x is the variable in question & signs depend upon last & middle terms signs (c is positive both will be the same as middle term, c is negative larger factor gets middle terms sign) Step 4: Check by multiplying Example: x 2 + 5x + 6 1) Factors of 6? 2) Which add to 5? 3) Write as a product of 2 binomials. Example: x 2 x 12 1) Factors of -12? 2) Which add to -1? 3) Write as a product of 2 binomials. Example: x 2 + x 12 1) Factors of -12? 2) Which add to 1? 3) Write as a product of 2 binomials. Example: x 2 + xy 2y 2 1) Factors of 2y 2? 2) Which add to 1y? 3) Write as a product of 2 binomials. Example: x 2 5x + 6 1) Factors of 6? 2) Which add to -5? 3) Write as a product of 2 binomials. Note: If 2 nd term and 3 rd term are both positive then factors are both positive. If 2 nd term and 3 rd are both negative or 2 nd term is positive and 3 rd term is negative then one factor is negative and one is positive. If the 2 nd term is negative and 3 rd is positive then both factors are negative. Y. Butterworth Ch. 5 Martin-Gay/Greene 22

23 Example: a 2 + 8a ) Factors of 15? 2) Which add to 8? 3) Write as a product of 2 binomials. Example: x 2 + x 6 1) Factors of -6? 2) Which add 1? 3) Write as a product of 2 binomials. Example: z 2 2z 15 1) Factors of -15? 2) Which add to -2? 3) Write as a product of 2 binomials. Example: x 2 17x ) Factors of 72? 2) Which add to -17? 3) Write as a product of 2 binomials. Example: x 2 3xy 4y 2 1) Factors of -4 y 2? 2) Which add to -3y? 3) Write as a product of 2 binomial Sometimes it is just not possible to factor a polynomial. In such a case the polynomial is called prime. This happens when none of the factors of the third term (constant usually) can add to be the 2 nd numeric coefficient. Example: x 2 7x + 5 If the leading coefficient (the first term in an ordered polynomial) is not one, try to factor out a constant first, then factor as usual. In this section, any time the leading coefficient is not 1, there will be a GCF, but that is not always true in the real world. If there is a variable common factor in all terms try to factor out that first. Example: Factor completely. a) 2x x + 12 b) 5x x 15 c) 7x 2 21x + 14 d) x 3 5x 2 + 6x Y. Butterworth Ch. 5 Martin-Gay/Greene 23

24 Sometimes the common factor is a more than a number and a variable, sometimes it is the product of several variable and sometimes it is even a binomial. Here are some examples. Example: Factor each completely. (Warning: Sometimes after you factor out the GCF you will be able to factor the remaining trinomial, and sometimes you won't.) a) (2c d)c 2 (2c d)c + 4(2c d) b) x 3 z x 2 z 2 6z 2 c) (a + b)a 2 + 4(a + b)a + 3(a + b) Another new idea in intermediate algebra is the concept of factoring using substitution. If we see a polynomial in quadratic form even though it may not look like a second degree polynomial, it can still be factored. There are two types of substitutions: 1) Higher degree than 2. a) Middle term s exponent, times 2 must give exponent of highest degreed term & there still must be a constant. i) Use middle term s variable as x in binomial Example: x 4 10x ) Binomial of descending degree a) 2 nd then 1 st then zero degree i) Use u to block out the binomial so you see u 2 ± u ± c and factor ii) Re-substitute binomial for u and simplify Example: (x + 2) 2 + 8(x + 2) + 15 Note: Don t confuse this example with a binomial GCF! The binomial GCF is same degree in every term. Y. Butterworth Ch. 5 Martin-Gay/Greene 24

25 Next we tackle the trinomials where the leading coefficient is not 1. These are the buggers! We are going to learn a trick first, and then we will come back to doing it the old fashioned way! Factoring a Trinomial by Grouping Step 1: Find the product of the 1 st and last numeric coefficients Step 2: Factor the product in one so that the sum of the factors is the 2 nd coefficient Step 3: Rewrite the trinomial as a four termed polynomial where the 2 nd term is now 2 terms that are the factors in step 2 Step 3: Factor by grouping Step 4: Rewrite as a product Example: 12x 2 11x + 2 1) Multiply the numeric coefficients of 1 st and last terms 2) Factors of number from step 1? (Hint: Use the prime factors of a &c to help you!) 3) Rewrite as the four termed polynomial where the middle terms are factor from step 2 that sum to 11 (note the middle term is negative so both must be negative) 4) Factor by grouping Example: 12x 2 + 7x 12 1) Multiply the numeric coefficients of 1 st and last terms 2) Factors of number from step 1? 3) Rewrite as the four termed polynomial where the middle terms are factor from step 2 that yield a difference of 7 (note the middle term is positive so the larger factor is positive and the other is negative) 4) Factor by grouping Example: 4x 2 9x 9 1) Multiply the numeric coefficients of 1 st and last terms 2) Factors of number from step 1? 3) Rewrite as the four termed polynomial where the middle terms are factor from step 2 that yield a difference of 9 (note the middle term is negative so the larger factor must be negative) 4) Factor by grouping Y. Butterworth Ch. 5 Martin-Gay/Greene 25

26 Example: 24x 2 58x + 9 1) Multiply the numeric coefficients of 1 st and last terms 2) Factors of number from step 1? 3) Rewrite as the four termed polynomial where the middle terms are factor from step 2 that yield a sum of 58 (note the middle term is negative so both must be negatives) 4) Factor by grouping Example: 12x x + 5 1) Multiply the numeric coefficients of 1 st and last terms 2) Factors of number from step 1? 3) Rewrite as the four termed polynomial where the middle terms are factor from step 2 that yield a sum of 17 4) Factor by grouping Example: 2x x ) Multiply the numeric coefficients of 1 st and last terms 2) Factors of number from step 1? 3) Rewrite as the four termed polynomial where the middle terms are factor from step 2 that yield a difference of 17 4) Factor by grouping Note: Sometimes these will be prime too! Example: 24x 2 y 2 42xy 2 + 9y 2 1) Factor out the GCF first 2) Multiply the numeric coefficients of 1 st and last terms 3) Factors of number from step 2? 4) Rewrite as the four termed polynomial where the middle terms are factor from step 3 that yield a sum of 14 5) Factor by grouping, don t forget to write the GCF Y. Butterworth Ch. 5 Martin-Gay/Greene 26

27 Example: 3(x + 2) 2 19(x + 2) + 6 1) Make your substitution first to create 3u 2 19u + 6 2) Multiply the numeric coefficients of 1 st and last terms 3) Factors of number from step 2? 4) Rewrite as the four termed polynomial where the middle terms are factor from step 3 that yield a sum of 19 5) Factor by grouping 6) Re-substitute (x + 2) and simplify Example: 14x 6 x 3 3 1) Make your substitution first to create 14u 2 u 3 2) Multiply the numeric coefficients of 1 st and last terms 3) Factors of number from step 2? 4) Rewrite as the four termed polynomial where the middle terms are factor from step 3 that yield a difference of 1 5) Factor by grouping 6) Re-substitute x 3 Now let s return to the traditional method. Don t be surprised if I skip this altogether and return to it in a few days. We will be using the same pattern as with x 2 + bx + c, but now we have an additional factor to look at, the first factor. Factoring Trinomials of Form ax 2 + bx + c Step 1: Find the factors of a Step 2: Find the factors of c Step 3: Find all products of factors of a & c (a 1 x + c 1 )(a 2 x + c 2 ) where a 1 x c 2 and c 1 a 2 x are the products that must add to make b! (This is the hard part!!!) The other choice is (a 1 x + c 2 )(a 2 x + c 1 ) where a 1 x c 1 and c 2 a 2 x must add to make b. And then of course there is the complication of the sign. Pay attention to the sign of b & c still to get your cues and then change your signs accordingly. (But you have to do this for every set of factors. You can narrow down your possibilities by thinking about your middle number and the products of the factors of a & c. If b is small, then the sum of the products must be small or the difference must be small and therefore the products will be close together. If b is large then the products that sum will be large, etc.) Step 4: Rewrite as a product. Step 5: Check by multiplying. (Especially important!) Y. Butterworth Ch. 5 Martin-Gay/Greene 27

28 Example: 2x 2 + 5x + 2 1) Look at the factors of the 1 st term 2) Look at the factors of the last term 3) Sum of product of 1 st and last factors that equal the middle term Ask yourself What plus what equals my 2 nd term? Lucky here that the 1 st and last terms are both prime that makes life very easy. Example: 10x 2 + 9x + 2 1) Factors of 10? 2) Factors of 2? 3) Product of factors that sum to 9? 9 is relatively small so we probably won t be multiplying 10 and 2! This eliminates at least one combination! Since 2 is prime and we know that 10 2 won t work that narrows our possibilities a lot! Example: 15x 2 4x 4 1) Factors of 15? 2) Factors of -4? 3) Product of factors that sum to -4? The difference is relatively small, so I won t be using 15 4 and 1 1 or 15 1 and 4 1, which actually eliminates quite a bit, since the only other factors of 4 are 2 and 2, which means that we just have to manipulate the sign. Sometimes when factoring the leading coefficient will be negative. It is easier to deal with problems that involve a negative coefficient if the negative is factored first and the focus can return to the numbers and not deal with unfamiliar signs. Example: Factor the following using techniques from this section and by factoring out a 1 first. -2a 2 5a 2 Y. Butterworth Ch. 5 Martin-Gay/Greene 28

29 Sometimes there will be a common factor in a trinomial, just like those found in binomials in section 3. This does not change what we must do, but after factoring out the binomial we must continue to look for factorization. Your Turn 1. Factor the following trinomials. a) x 2 + 3x + 2 b) y 2 + 2y 15 c) z 2 12z 28 d) r 2 7r Factor each trinomial completely. a) 9x 2 18x 27 b) 2x 2 y + 6xy 4y c) (2a + 1)a 2 5(2a + 1)a 6(2a + 1) d) (2z + 1)z 2 4(2z + 1)z 6(2z + 1) 3. Factor each of the following trinomials completely. a) 2x 2 + 5x 3 b) 12x 2 + 7x + 1 Y. Butterworth Ch. 5 Martin-Gay/Greene 29

30 4. Factor the following completely. a) 15x 2 (r + 3) 34x(r + 3) 16(r + 3) b) 21x 2 48x 45 c) 6x 2 y + 34xy 84y 5. Factor each of the following using factoring by grouping. a) 10x 2 13xy 3y 2 b) 72x 2 127x + 56 Y. Butterworth Ch. 5 Martin-Gay/Greene 30

31 5.7 Special Products Sometimes we will see some of the special patterns that we talked about in 5.3, such as: a 2 + 2ab + b 2 = (a + b) 2 or a 2 2ab + b 2 = (a + b) 2 These are perfect square trinomial. They can be factored in the same way that we've been discussing or they can be factored quite easily by recognizing their pattern. Factoring a Perfect Square Trinomial Step 1: The numeric coefficient of the 1 st term is a perfect square i.e. 1,4,9,16,25,36,49,64,81,100,121,169,225, 256 etc. Step 2: The last term is a perfect square Step 3: The numeric coefficient of the 2 nd term is twice the product of the 1 st and last terms' coefficients square roots Step 4: Rewrite as: ( 1 st term + last term ) 2 or ( 1 st term - last term ) 2 Note: If the middle term is negative then it's the difference of two perfect squares and if it is positive then it is the sum. Note2: that whenever we see the perfect square trinomial, the last term is always positive, so if the last term is negative don't even try to look for this pattern!! Example: x 2 + 6x + 9 1) Square root of 1 st term? 2) Square root of last term? 3) Twice numbers in two and three? 4) Factor, writing as a square of a binomial Example: 4x 2 12x + 9 1) Square root of 1 st term? 2) Square root of last term? 3) Twice numbers in two and three? 4) Factor, writing as a square of a binomial Example: 32x x ) GCF 1 st 2) Square root of 1 st term? 3) Square root of last term? 4) Twice numbers in two and three? 5) Factor, writing as a square of a binomial Y. Butterworth Ch. 5 Martin-Gay/Greene 31

32 Difference of Two Perfect Squares Remember the pattern: (a + b)(a b) = a 2 b 2 Example: (x 3)(x + 3) = x 2 9 Now we are going to be "undoing" this pattern. Factoring the Difference of Two Perfect Squares Step 1: Look for a difference binomial and check a) Is there a GCF? If so, factor it out and proceed with b) & c) b) Is 1 st term coefficient is a perfect square? (If no, then stop, problem is complete) c) Is 2 nd term is a perfect square? (If no, then stop, problem is complete) Step 2: Yes to both b) and c) then factor the difference binomial in the following way ( 1 st term + 2 nd term) ( 1 st term 2 nd term) Step 3: If there was a GCF don t forget to multiply by that GCF. Example: Factor completely. a) x 2 y 2 b) 4x 2 81 c) z 2 1 / 16 d) 27z 2 3y 2 e) 12x 2 18y 2 Note: Sometimes there is a common factor that must be factored 1 st & sometimes after factoring a GCF, the remaining binomial can t be factored. Any time an exponent is evenly divisible by 2 it is a perfect square. If it is a perfect cube it is evenly divisible by 3, and so forth. So, in order to factor a perfect square binomial that doesn t have a variable term that is square you need to divide the exponent by 2 and you have taken its square root. Example: Factor completely. a) 16x 4 81 b) 9x 6 y 2 Note: Sometimes the terms can be rewritten in such a way to see them as perfect squares. In order to see x 4 as a perfect square, think of (x 2 ) 2. Sometimes we will need to use our factoring by grouping skills to factor special binomials as well as just a GCF. Here is an example. Example: Factor the following completely. 3x 2 3y 2 + 5x 5y Y. Butterworth Ch. 5 Martin-Gay/Greene 32

33 The SUM of Two Perfect Squares a 2 + b 2 is PRIME unless there is a GCF Example: Factor completely. a) 25x y 2 b) 25x Note: Part a is prime, and part b only factors because it has a GCF. Remember that even when the polynomial you create is prime, if you ve created a product, you have factored and you do not write prime. Sum and Difference of Two Perfect Cubes This is the third pattern in this section. The pattern is much like the pattern of the difference of two perfect squares but this time it is either the sum or the difference of two perfect cubes. At this time it might be appropriate to review the concept of a perfect cube and or finding the cube root of a number. It may also be appropriate to review some perfect cubes: 1 3 =1, 2 3 =8, 3 3 =27, 4 3 =64, 5 3 =125, 10 3 =1000 If you have the difference of two perfect cubes then they factor as follows: a 3 b 3 = (a b)(a 2 + ab + b 2 ) If you have the sum of two perfect cubes then they factor as follows: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) Factoring the Sum/Difference of Two Perfect Cubes Step 1: Look for a sum or difference binomial and check a) Is there a GCF? If so, factor it out and proceed with b) & c) b) Is 1 st term coefficient is a perfect cube? (If no, then stop, problem is complete) c) Is 2 nd term is a perfect cube? (If no, then stop, problem is complete) Step 2: Yes to both b) and c) then factor the difference binomial in the following way Where a = cube root of the 1 st term and b = the cube root of the 2 nd term If the binomial is the difference (a b)(a 2 + ab + b 2 ) If the binomial is the sum (a + b)(a 2 ab + b 2 ) Step 3: If there was a GCF don t forget to multiply by that GCF. Note: In the factored form, the binomial follows the binomial s sign, and the middle term of the trinomial is the opposite of the binomials sign, and the sign of the last term is always positive. Y. Butterworth Ch. 5 Martin-Gay/Greene 33

34 Example: Factor each of the following perfect cube binomials. a) 125x b) 27b 3 a 3 c) 24z d) 48x 3 54y 3 e) x f) x 3 y 6 64 g) (x + 1) 3 8 h) (2x 1) One last type of special factoring for intermediate algebra: A difference of a perfect square trinomial and a perfect square. This one is tricky because it is again a 4 termed polynomial, and it is the first time we will have seen this pattern so we won t be expecting it. Here s what to look for: 1) 4 terms 2) 3 perfect squares, two of the same sign, and one of the opposite a) Usually 2 positive & one negative Here s what to do: 1) Separate perfect square trinomial & perfect square 2) Factor perfect square trinomial 3) Factor difference of 2 perfect squares using substitution 4) Re-substitute and simplify Example: Factor a) a 2 + 2a + 1 b 2 b) x 2 18x + 81 y 2 Warning: Integrated Review follows this section. Homework due next class! Y. Butterworth Ch. 5 Martin-Gay/Greene 34

35 5.8 Solving Equations by Factoring and Problem Solving A quadratic equation is any polynomial which is a 2 nd degree polynomial and is set equal to zero. A quadratic is said to have the form: where a, b, c are real numbers and a 0. ax 2 + bx + c = 0 A quadratic equation can be written other than in the above form, which is called standard form, but it can always be put into standard form. To get a quadratic in standard form, simply use the addition property of equality to move all terms to one side of the equal sign. It is best to keep the leading coefficient positive. Let's practice. Example: Put the following into standard form. a) x 2 2x = 5 b) 2x + 5 = x 2 2 c) 15 2x = x 2 5x + 3 BTW: A polynomial function where P(x) = 0 is a quadratic in standard form. This fact will come into play in a while. Our next task is solving a quadratic equation. Just as with any algebraic equation such as x + 5 = 0, we will be able to say that x = something. This time however, x will not have just one solution, it will have up to two solutions!! In order to solve quadratics we must factor them! This is one reason why we learned to factor. There is a property called the zero factor property that allows us to factor a quadratic, set those factors equal to zero and find the solutions to a quadratic equation. The zero factor property is based upon the multiplication property of zero anything times zero is zero. Thus if one of the factors of a quadratic is zero then the whole thing is zero and that is the setup! Y. Butterworth Ch. 5 Martin-Gay/Greene 35

36 Solving Quadratic Equations Step 1: Put the equation in standard form Step 2: Factor the polynomial Step 3: Set each term that contains a variable equal to zero and solve for the variable Step 4: Write the solution as: variable = or variable = Step 5: Check Sometimes book exercises give you equations where step 1 or steps 1 and 2 have already been done. Don t let this fool you, the steps from there on are the same. Example: Solve the following: a) x 2 4x = 0 b) x 2 6x = 16 c) x 2 = 4x + 3 d) -2 = -27x 2 3x Sometimes it is necessary to multiply a factor out in order to arrive at the problem in standard form. You will realize that this is necessary when you see an equation that has one side that is factored but those factors are equal to some number or when there are sums of squared binomials or squared binomials that equal numbers. e) (2x 5)(x + 2) = 9x + 2 f) a 2 + (a + 1) 2 = -a f) y(2y 10) = 12 There is also the case where we have a greatest common factor or which can be solved by factoring by grouping. These all use the same principles. Example: Solve each of the following by applying the zero factor property to give the solution(s). a) 3x 3 + 5x 2 2x = 0 b) 4y 3 = 4y 2 + 3y Y. Butterworth Ch. 5 Martin-Gay/Greene 36

37 c) (2x + 1)(6x 2 5x 4) = 0 d) 125x 2 = 245 X-Intercepts of a Parabola We already know that y = ax 2 + bx + c can be graphed and that the graph of a quadratic equation in two variables is a parabola. We have also discussed that this equation will be increasing or decreasing according to the sign of a. We know that it has a vertex and is symmetric [vertex is at (- b / 2a, f(- b / 2a ))] about the line of symmetry which goes through this vertex [and is therefore the line x = - b / 2a ]. What we have not discussed is that just as with linear equations, parabolas also have intercepts. Recall that an x-intercept is a place where the graph crosses the x-axis. Lines only do this at one point, but because of the nature of a parabola, it is possible for this to happen twice (vertex is below x-axis and parabola opens up, or vertex is above the x-axis and the parabola opens down), once [vertex is on x-axis] or even never (vertex is above the x-axis). Just as with lines, the x-intercept is found by letting y = 0 and solving for x. This gives us our relationship to quadratic equations, when we look at a quadratic as a polynomial function, and set the function (the y-value) equal to zero P(x) = 0. Finding X-Intercept(s) of a Parabola Step 1: Let y = 0, if no y is apparent, set the quadratic equal to zero or let P(x) = 0 Step 2: Use skills for solving a quadratic to find x-intercept(s) Step 3: Write them as ordered pairs (remember there could be one or two or even none) Example: Find the x-intercept(s) for the following parabolas a) y = (2x + 1)(x 1) b) y = x 2 + 2x + 1 c) y = x 2 4 Pythagorean Theorem The Pythagorean Theorem deals with the length of the sides of a right triangle. The two sides that form the right angle are called the legs and are referred to as a and b. The side opposite the right angle is called the hypotenuse and is referred to as c. The Pythagorean Theorem gives us the capability of finding the length of one of the sides when the other two lengths are known. Solving the Pythagorean theorem for the missing side can do this. One of the legs of a right triangle can be found if you know the equation: Y. Butterworth Ch. 5 Martin-Gay/Greene 37

38 Pythagorean Theorem a 2 + b 2 = c 2 Solving the Pythagorean Theorem Step 1: Substitute the values for the known sides into the equation Note: a is a leg, b is a leg and c is the hypotenuse Step 2: Square the values for the sides Step 3: Solve using methods for solving quadratics or using principles of square roots Example: One leg of a right triangle is 7 ft. shorter than the other. The length of the hypotenuse is 13 ft. Find the lengths of the legs. Example: Two cyclists, Tim and Joe, start at the same point. Tim rides west and Joe rides north. At some time they are 13 miles apart. If Tim traveled 7 miles farther than Joe, how far did each travel? Geometry Problems Geometry problems are problems that deal with dimensions, so always remember that negative answers are not valid. As with number problems (as encountered in Algebra) it is possible to get more than one set of answers. Geometry problems that we will encounter will deal with the area of figures and the Pythagorean Theorem. We've just discussed the Pythagorean Theorem. Example: Find the dimensions of a rectangle whose length is twice its width plus 8. Its area is 10 square inches. Y. Butterworth Ch. 5 Martin-Gay/Greene 38