(8m 2 5m + 2) - (-10m 2 +7m 6) (8m 2 5m + 2) + (+10m 2-7m + 6)

Transcription

1 Adding Polynomials Adding & Subtracting Polynomials (Combining Like Terms) Subtracting Polynomials (if your nd polynomial is inside a set of parentheses). (x 8x + ) + (-x -x 7) FIRST, Identify the like terms. Like terms have matching bases that have matching exponents. (x 8x + ) + (-x - x 7) (8m m + ) - (-0m +7m 6) (8m m + ) - (-0m +7m 6) Before you do anything else, DISTRIBUTE THE SUBTRACTION SIGN. (THINK OF IT AS AN OPPOSITE SYMBOL) x 8x + + -x - x 7 -x - x Final answer THEN, Add the coefficient. Yes, you still need to know your integer rules. BUT, Keep the matching bases AND Keep the matching exponents. (8m m + ) + (+0m -7m + 6) (8m m + ) + (+0m -7m + 6) 8m m + + 0m -7m + 6 8m m + 8 Final answer Now, just follow the adding rules

2 Multiplying Powers With the Same Base Keep the common base. Add the exponents LAWS OF EXPONENTS: work for Multiplication & Division Dividing Powers With the Same Base Keep the common base. Subtract the exponents. 8 = 8- = Raising a Power To Another Power Keep the common base. Multiply the exponents. ( ) 6 = 6 = 8 Raising a Fraction To a Power Raise the numerator to the power. Raise the denominator to the power. ( ) = = 9 Raising a Product To a Power Raise EACH FACTOR to the power. ( ) = = 9 6 Positive Exponents Negative Exponents Zero Exponents is what you start with. Then, it is repeatedly being DIVIDED by the base. is what you start with. Then, it is repeatedly being MULTIPLIED by the base. is what you start with. NOTHING IS DONE TO IT. So, is what you end up with. + + ( multiplied by times) - ( multiplied by times) 0 ( is NOT MULTIPLIED or DIVIDED by at all!) Any Base Raised to the 0 power is equivalent to. EXCEPTION to the Rule 0 0 is undetermined. Check it out in your calculator,

3 } } } LAWS OF EXPONENTS: Special cases with signed coefficients Positive Coefficient (with or without parentheses) Negative Lead Coefficient without parentheses Negative Lead Coefficient with parentheses POSITIVE is what you start with. Then, it is repeatedly being Multiplied by the base of +. NEGATIVE is what you start with. Then, it is repeatedly being Multiplied by the base of +. POSITIVE is what you start with. Then, it is repeatedly being Multiplied by the base of (-) (-) NOTE: If the negative sign is inside the parentheses and the exponent is an even number, the answer will be positive (as shown above). If the exponent is an odd number, then the answer will be negative.

4 When Final Answers Require MORE Simplification. Final Answers Should Have Positive Exponents Answers that have Zero Exponents When your answer results in a zero exponent, remember to finalize your answer. It will equal. Ex: = -- = 0 = Answers that have Negative Exponents When your answer results in a negative exponent, remember to finalize your answer. Convert to an equivalent fractional form (division problem) with a positive exponent. = - = - = NEGATIVE exponents in the denominator should be simplified further as well! Reverse the numerator and the denominator. Then, make the exponent positive. Ex: 8 = 8

5 How Can 0 be equivalent to???? This is an Alternate Way To Think About It, That Might Help You Believe It. Here is what it looks like expanded out: Ex: = But we know that If = = is also equal to -- = 0 = 0 and = Then, 0 must equal. How Can - be equivalent to???? This is an Alternate Way To Think About It, That Might Help You Believe It. Here is what it looks like expanded out: Ex: = But we know that If = = - and is also equal to - = - = = Then, - must equal

6 Multiplying & Dividing Monomials Part Old-Fashioned Multiplying & Dividing Part Laws of Exponents Multiplying Monomials Dividing Monomials 0 (a b c 7 )(-a b c ) The Commutative Property allows us to rearrange a multiplication problem to a form to works better for us. Coefficients near Coefficients Like Bases near Like Bases -a a b b c 7 c Divide the coefficients just as you would any normal numbers. 0 0a a b b 7 b b c 6 c Think of this as several little division problems. a a 0 7 c c 6 To divide same bases, you subtract the exponents. Multiply the coefficients just as you would any normal numbers. Yes you will need to know your integer rules. -a b 8 c 9 Final simplified answer To multiply same bases, you add the exponents. Yes you will need to know your integer rules. a b c - a b c Final simplified answer Remember, when you have negative exponents in your final answer, you need to rewrite using positive exponents in your final step. c - really means repeated division by c. 6

7 Example: Multiply x (x + x - ) To Multiply Polynomials just DISTRIBUTE: x (x + x - ) Multiply x by each term inside the parentheses. (x x ) + (x x ) + ( x -) Answer: x + x - 6x Dividing POLYNOMIALS by Monomials: ) Divide each term of the polynomial by the monomial. ) Hints about your answers: * If you are dividing a trinomial, your answer will be a trinomial. * If you are dividing a binomial, your answer will be a binomial. * If you are dividing a monomial, your answer will be a monomial. Example: Divide (x + x - 6x ) (x ) The other way to write a division problem. x + x - 6x Notice the long fraction bar means all terms x are being dividing by x. x x + My Division Work x 6x - x x x + x - Answer 7

8 Expressions VS Polynomial Expressions Expression: Numbers, symbols and operations (such as +, -, and ) grouped together that show the value of something. A) Term of an expression: A number, a variable, or a product or quotient of numbers and variables. Notice single terms do not contain addition or subtraction. q ab x - xyz Terms are added/subtracted in algebra expressions. Each time you encounter an addition/subtraction symbol this is how you know that you are starting a new term. Example : ab + Example : ab - ( ab is the st term and is the nd term. ) ( ab is the st term and - is the nd term. ) The examples above are both considered to be expressions with two terms. Terms can be categorized as Like terms or Unlike terms. ) Like terms: Terms that contain exactly the same variables, with each matching variable containing the same exact exponent. Examples: and 7 (All constants are considered like terms.) y and y 9ab and ab x y and x y ) Unlike terms: Terms that contain different variables or terms that have the same variables, but different exponents. Examples: x and y 9ab and a x y and x y B) Coefficient of a term: A number before a variable. This constant tells how many of a variable you have. Ex: a really means a + a + a + a + a NOTE: Like terms can have different coefficients, but the variable/exponent combinations must match exactly. In the example x y and x y listed above, the x y represents the same quantity in both terms. There is set of x y and another sets of x y. Therefore, it makes logical sense to combine them and just say there are sets of x y. C) Fully Simplified Expressions: In order to be simplified an expression must only contain monomial terms, separated by + or signs. There cannot be any. Like terms that still need to be combined by addition or subtractions. Example: a + a is not simplified, until you combine like terms to 9a. Laws of exponents that can still be applied (multiplying same bases, dividing same bases, power to power, changing an x 0 to, and converting negative exponents.) Example: a b a is not fully simplified, because the a s can be joined by adding the exponents. The expression simplifies to a 7 b. Factors that have not been distributed. Example: 8x(x ) is not simplified fully. The 8x is a monomial factor and the x - is a binomial factor. The 8x needs to be distributed (multiplied) into the x -, to simplify the expression to x 0x. w 8

9 Polynomial Expression: Numbers, symbols and operators (such as +, -, and ) grouped together that show the value of something. It is almost the same as a basic expression, but with one exception In simplified form, Polynomials expressions cannot contain division by a variable, even though a basic expression can. Polynomial Term OR Polynomial Expression q ab x - xyz Basic Term OR Basic Expression (division by a variable) w a b a - Remember: a - = a, which means you are dividing by a variable. So, it is not a polynomial. A) Categorizing Polynomials: A polynomial can be categorized by the number of unlike terms it contains. Monomial Binomial Trinomial (One Term) (Two Unlike Terms) (Three Unlike Terms) 7x + x + c + 7 x 8x + 7y a + b + c 9z 7a + bc x + xy + y Terms vs Factors in an Expression: Terms: Remember that individual terms are numbers, variables, or a product or quotient of numbers and variables, where each new term is separated by a (+ or -). Factors: Numbers that are multiplied together. Ex) ( and are the factors, since they are being multiplied.) Expressions that are multiplied together. Ex) a(a - ) (The factors are the monomial a and the binomial a since these expression are being multiplied.) x y + 8x y Can be classified as a binomial expression. Contains terms, x y and 8x y. The term x y has factors of, x, and y. The term 8x y, has factors of 8, x, and y. More examples of terms vs. factors (x + )(x + 8x - ) This is a polynomial, but it is not simplified, because it is in factored form. The factors are a binomial x + and a trinomial x + 8x. The factor x + has terms x and. The term x in the factor x - has it s own factors of and x. Etc, etc 9

10 Polynomials: Standard Form and Degree A) Polynomials with ONE Variable (Degree and Standard Form) ) The degree corresponds to term with highest exponent on a variable. Example : x + 7x (which equals x + 7x x 0 ) The degree of this polynomial is. ) Standard Form: The exponents in the terms of the polynomial should be in descending order. Example : -x + x + x becomes x - x + x B) Polynomials with MORE than one Variable (Degree and Standard Form) ) The degree corresponds to the term in which the sum of the exponents on the variables is the largest. Example y z + x z 9-8x y 6 xy Sum of exponents Sum of exponents Sum of exponents Sum of exponents The degree of this polynomial is 0. ) Standard Form: The sum of the exponents in the terms of polynomial should be in descending order. If there are terms with the same degree, then the term that has a variable that comes first in the alphabet should come first. Example : y z + x z 9-8x y 6 xy Sum of exponents Sum of exponents Sum of exponents Sum of exponents becomes x z 9-8x y 6 + y z xy Sum of exponents Sum of exponents Sum of exponents Sum of exponents Both terms have a degree of 8, but, x comes before y in the alphabet, so 8x y 6 is written st. Reminder: A polynomial should be fully simplified before placing it in standard form. (see your prior notes for a reminder). 0

11 DOUBLE DISTRIBUTE: Example: Multiply (x )(x + ) st, set up single distribution expressions. nd, distribute the x into x -. rd, distribute the - into x -. th, combine any like terms created. x(x + ) + -(x + ) 6x + x + -0x 0 FINAL Answer 6x + x 0 Alternate Technique FIRST (F): Multiply the FIRST TERMS in each pair of binomials. F.O.I.L. Technique OUTER (O): Multiply the OUTER TERMS in each pair of binomials. INNER (I): Multiply the INNER TERMS in each pair of binomials. LASTS(L): Multiply the LAST TERMS in each pair of binomials. FIRSTS 6x x ) (x + ) INNER -0x OUTER x LASTS -0 Combine Like Terms to get the final answer: 6x + x -0x - 0 6x + x 0

12 Greatest Common Factor (GCF) : The greatest common factor of or more numbers is the largest factor the numbers have in common. The GCF of and 8 is 6. 6 = 6 = 8 Although they have other common factors, 6 is the largest. The greatest common factor of or more powers is the largest power each base has in common. The GCF of x and x is x. x x x x x = x x x x = x Notice when you expand each power they have factors of x in common, which is x and this is your GCF. You will be asked to find the GFC for terms that have a combination of coefficients (numbers) and powers. Find the GCF of each part separately, then put each part together to create the complete GCF. Example: Find the GCF of x and 8x The GCF of & 8 is 6 The GCF of x & x is x So, the GCF of x and 8x is 6x Find the GCF of 0m 6 and 0m 8 The GCF of 0 and 0 is 0 The GCF of m 6 and m 8 is m 6 So, the GCF of 0m 6 and 0m 8 is 0 m 6 NOTICE: In each case the GCF was the base with the smallest power. For example: For x & x the GCF was x. For m 6 & m 8 the GCF was m 6..

13 Observations When Double Distributing That Help You Understand Reverse Distribution (Factoring) (x )(x + ) x(x + ) + -(x + ) 6x + 9x + -x - -8x Product of the Inner Terms -8x Product of the Outer Terms Notice: If you check mid-way through the process, you will notice that The product of the inners EQUALS The product of the outers 6x + 9x + -x - 7x Sum of the Inner Terms 6x + 7x - Inner Term in the Final Trinomial is also 7x Notice: If you check at the end of the process, you will notice that The sum of the inners EQUALS The sum of the final middle term of the trinomial CONCLUSION: To reverse distribution (factor a trinomial) You must find a pair of unique numbers. The numbers must Have a product that is equal the product of the outer terms of the trinomial. But the same numbers must also, Have a sum that is equal to the middle term of the trinomial. WE WILL PRACTICE THIS TECHNIQUE IN THE NEXT LESSON

14 Reversing DOUBLE DISTRIBUTION: FACTORING a Quadratic Expression by. st : Determine the product of the a & c coefficients. (-0). Identify the b coefficient (). nd : Determine numbers that have a product of -0. But, the same pair of numbers must also have a sum of ONLY UNIQUE PAIR WILL WORK. In this case, that pair is - &. Finding the PRODUCT / SUM To SPLIT THE MIDDLE Product -0 - Sum rd : Split the b-term x into terms using the pair of numbers you just found. The term x will become -x and +x. ax + bx + c Product of -0 Sum of 0x + x Split x into -x and +x, using the - &, since it has the correct product/sum. You will now have terms instead of. 0x - x + x th : Factor the GCF from first half of the expression. Factor the GCF from last half of the expression. Factor by GCF Factor by GCF x(x - ) + (x - ) th : Rewrite the expression as Binomials. The GCF s that you factored out form the first binominal. The common binomial that remains forms the nd binomial. Create a binomial, using the GCF s of x &. (x + )(x - ) FINAL ANSWER Create a nd binomial, by using the common binomial x -.

15 CHECKING YOUR ANSWER BY HAND Since Factoring is the reverse of distribution, Distribution is the reverse of factoring! So, check your Factored answer with Distribution. (x + )(x - ) Double Distribution-To Check x(x ) + (x ) 0x x + x 0x + x CHECKING YOUR ANSWER USING THE TI-8 The objective is to ensure the original expression is EQUIVALENT to the new expression. If the expressions are equivalent, then placing both in the TI-8, as equations, SHOULD result in identical input(x)/output(y) charts. It also, SHOULD result in identical graphed images. TO CHECK: ) Press. Place both expressions in your calculator. ) Your screen should look like this when you are done. HINTS: For a variable x press Before you key in your exponent press To move out of exponent mode press ) Press to make sure the GRAPHS match. 0x + x Since this is the original Quadratric Expression (before factoring), then our factoring is correct. Press The images are the same, so it looks like only is graphed. to make sure Y and Y match. The Y and Y columns are identical, which implies the expressions are equivalent..

16 x -x -0 FACTORING When the Lead a Coefficient is. Remember there is always an invisible Lead coefficient if no number is written. SHORTCUT RULE When the lead a coefficient is the product/sum values - & are the same values in your final answer. This allows you to SKIP all the Split the Middle steps. Go straight to your final answer and create your factors using the product/sum numbers. BUT, BE CAREFUL!!! This shortcut only works when the lead coefficient is. Product of -0 x -x - 0 Sum of - Product -0 - Sum - (x + )( x - ) Proceed Straight to the Final Answer Use the Product/Sum Values to Create the Factors NOTICE: Split the Middle is UNNECESSARY if your Lead Coefficient is. SO, SKIP all the Split the Middle Step!!! x -x 0 x - x + x 0 x(x - ) + (x - ) (x + )( x - ) 6

17 Special Cases Factoring by Split the Middle Factoring by Split the Middle for a Degree Trinomials Solve just as you would with a Degree Polynomial, except You will be splitting up x s instead of x s. Your binomial factors with have x s instead of x s. Degree Trinomial Ex: x + 9x + 8 x + x + 6x + 8 x (x + ) +6(x + ) (x + 6) (x + ) Degree Trinomial Ex: x + 9x + 8 x + x + 6x + 8 x (x + ) +6(x + ) (x + 6) (x + ) 7

18 Difference of Perfect Squares (DOPS) In this case there is a binomial with the following characteristics. BOTH terms are perfect squares. The terms are separated by a subtraction sign. You have Options. Learn both. Each method will come in handy! OPTION : Create a trinomial by placing a 0x into the binomial to create the middle term of the trinomial. Then, solve as you usually would using product/sum AND split the middle. OPTION : Square Root the first term x. Use the square x as the first term in each factored binomial. Square Root the send term. Use the square root of as the nd term of each factored binomial. Place a + in the first binomial. Place a in the nd binomial. Ex: x - x +0x - x + x -x - x(x + ) -(x + ) (x + ) (x - ) Ex: x - Square root of x is x. Square root of is. THE ANSWER is (x + ) (x - ) 8

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