Math 154 :: Elementary Algebra

Transcription

1 Math 1 :: Elementar Algebra Section.1 Exponents Section. Negative Exponents Section. Polnomials Section. Addition and Subtraction of Polnomials Section. Multiplication of Polnomials Section. Division of Polnomials Answers

2 Math 1 :: Elementar Algebra Section.1 Exponents Examples: Simplif each expression. a) In this expression, is the base and is the exponent. The parentheses around include the negative sign in the base. To simplif: b) In this expression, is the base and is the exponent. There is a multiplication b 1 as well. There are no parentheses, so the negative sign is not part of the base. The expression ma be written as: 1 To simplif: 1 c) 1 78x 10 x 1 78x 10 x 1 x 10 x 9 x 9 x 9x In this expression, it is easiest to simplif the inside of the parentheses before appling the outside exponent. First, simplif the fraction 78 b dividing into both 78 and. Next, cancel three factors of x from the numerator and denominator, and then cancel five factors of from the numerator and denominator. It ma help to visualize the cancellation like this: x x x x x x x x x x x x x x x Now, appl the outside exponent to each factor in the numerator and denominator, b squaring and, and multipling the exponents on x and. Homework 1. When multipling like-bases, what operation can ou perform on the exponents to simplif the expression?. When dividing like-bases, what operation can ou perform on the exponents to simplif the expression?. How do ou determine if the factors in a simplification of a quotient of like-bases are in the numerator or the denominator?. When raising a power to a power, what operation can ou perform on the exponents to simplif the expression? Section.1 Exponents 1

3 Math 1 :: Elementar Algebra. When simplifing a quotient that is raised to a power, what is often the easiest step to take first?. If an expression is raised to the 0 power, what is its value? 7. Simplif each expression. a) b) c) d) e) 8. Simplif each expression. a) b) c) d) e) Simplif. 9. xx z z x d d xx z 1 z x x. Section.1 Exponents

4 Math 1 :: Elementar Algebra.... c d c d 9 x x x x 8 7. x w 1 w x x x x x 0. x. x x. x cd 1 c d 0a 1 0a x 0x m 1m 0 n 0 1 n p q 8 pq x m n Section.1 Exponents

5 Math 1 :: Elementar Algebra x 8x p q p q 1x 1x 0 9m m ac 9 1 ac 9. x x m m. x x x 1x n n. z z Section.1 Exponents

6 Math 1 :: Elementar Algebra Section. Negative Exponents Examples: Simplif each expression. a) In this expression, is the base and is the exponent. The parentheses around include the negative sign in the base. The negative exponent moves the factors to the denominator. To simplif: 1 1 b) In this expression, is the base and is the exponent. There is a multiplication b 1 as well. There are no parentheses, so the negative sign is not part of the base. The negative exponent moves the factors to the denominator. The expression ma be written as: 1 To simplif: c) 8 1x x 8 1x x In this expression, it is easiest to move the factors with negative exponents first. That means that and move to the numerator, and moves to the denominator. The exponents on these factors become their opposites during the move. x 8 x x 8 1x x Now, simplif the fraction 1 to. Next, combine the factors of x in the numerator b adding the exponents and cancel three factors of from the numerator and denominator. x 1 Homework 1. In our own words, describe what happens to the exponent of a factor when the factor is moved from the numerator to the denominator of a fraction?. In our own words, describe what happens to the exponent of a factor when the factor is moved from the numerator to the denominator of a fraction? Section. Negative Exponents

7 Math 1 :: Elementar Algebra. Simplif each expression. a) b) c) d) e). Simplif each expression. a) b) c) d) e) Simplif. Final answers should not contain negative exponents.. x x x m n x 8 7 x 1. z 8 z x d d xx z 10 z Section. Negative Exponents

8 Math 1 :: Elementar Algebra x x x x 7 9. x w w 1x 10 x 18 x x x x 0. x x 1. x cd c 1 0a 0a x 0x m 1m 0 p p x d n 1 n 10 8 q 8 q m 7 n 8x x 1 Section. Negative Exponents 7

9 Math 1 :: Elementar Algebra p q 8p q 1x x 1 7m m a c 9 1 ac. x x m m 9. x x x x n n Section. Negative Exponents 8

10 Math 1 :: Elementar Algebra Section. Polnomials. Polnomials Worksheet Example: For the polnomial given, find the degree of each term, the degree of the polnomial, the leading term, and the leading coefficient. If the polnomial has a specific name monomial, binomial, or trinomial give that name. a) 9x x 1 Individual Terms The Degree of Each Individual Term The Coefficient of Each Individual Term The Leading Coefficient of the Polnomial The Degree of the Polnomial Specific Name of the Polnomial 9x 7 9 x trinomial Homework 1. In our own words, define a polnomial.. In our own words, define each word: monomial, binomial, trinomial.. In our own words, describe how ou identif the degree of a polnomial.. In our own words, describe how ou identif the leading term of a polnomial.. In our own words, define each word: constant, linear, quadratic, cubic. For each polnomial given, find the degree of each term, the degree of the polnomial, the leading term, and the leading coefficient. If the polnomial has a specific name monomial, binomial, or trinomial give that name. You ma use a chart like the one below for each polnomial, but it isn t necessar, as long as ou identif each answer. Individual Terms The Degree of Each Individual Term The Coefficient of Each Individual Term The Leading Coefficient of the Polnomial The Degree of the Polnomial Specific Name of the Polnomial x x 11 x x x 17 1x x 7 1 Arrange each polnomial in descending order. Give the degree of each polnomial and the leading coefficient x 7x 1x x Section. Polnomials 9

11 Math 1 :: Elementar Algebra Section. Addition and Subtraction of Polnomials Examples: Perform the operation. Answers ma be written in descending order of power, but it isn t necessar. a) 7x 1x x x To add two polnomials, combine like-terms. Exponents on variables will NOT change. If terms are reordered, take the sign in front of the term with the term that s moved. 7 x 1 x 1 x x 7x x 1x x 8x 7x b) Subtract x 0x from 11x x Homework When translating a statement that involves subtract from, polnomials are switched between the math order and the English order. This problem becomes: 11x x x 0x 11x x x 0x Distribute the subtraction sign to all of the terms in the parentheses following it. 1x 1x 9 Combine like-terms. 1. In our own words, describe a like-term. What must be the same? What ma be different?. In our own words, describe how to add two polnomials. What changes? What doesn t change? Perform the operation. Answers ma be written in descending order of power, but it isn t necessar.. x x. 1. xx x x x x x 9. 7 x 8x 10. 1x 8x x x 11. x8 11x 1. m 0 m m 1. m 8m 9 m a a 1 a x 7x x 1x Section. Addition and Subtraction of Polnomials 10

12 Math 1 :: Elementar Algebra 17. x1 x x1 8x 19. m 18 m m 0. n 10n 11 n a 7a 8 a 1.. x 0x 8 x 1x. 9x x m m m 7. x 18 x x 7. a a 1 a Add x and x x Add 1 and. 0. Subtract 9x from x Subtract 8 from 1.. Subtract m m 18 from m 11m 9.. Subtract x 7x19 from x x. Section. Addition and Subtraction of Polnomials 11

13 Math 1 :: Elementar Algebra Section. Multiplication of Polnomials Examples: Perform each operation. Simplif answers (if not simplified after multipling). a) 1 7 To multipl a monomial b a polnomial, distribute the monomial to each term in the polnomial. Exponents on variables MAY change. 1 7 b) x x To multipl two binomials, ou ma use the FOIL method. FOIL stands for the multiplications of the terms: First, Outer, Inner, and Last. FOIL-ing is the same thing as distributing each term in the first polnomial to each term in the second polnomial. x x x x x x x 10x x x 7x c) x To square a binomial, multipl it out using the FOIL method. There is a pattern. If ou recognize it, ou are welcome to use it. x x x x x x x x x x 1 x 8x1 Homework 1. What propert is most used when multipling polnomials?. When computing the square of a binomial for example, an expression of the form. Compute each problem. a) x b) x c) x d) x a b what must ou remember? e) Using the above problems as examples, in our own words, describe how ou can tell when ou ma use a shortcut exponent rule and when ou must FOIL? Perform each operation. Simplif answers (if not simplified after multipling). x 7.. x x 8. x x 1 9. xx Section. Multiplication of Polnomials 1

14 Math 1 :: Elementar Algebra x x x x x x x 7x 1. mn m mn dd xx k k 0. p p 1. x1 x x 7. xx 9. k 1k x 8. p 1 p 8 9. x 1x 0. k11k x 1. mp 1mp 10. x x 7. a ca c. x x. x 9 7. x x 8. x Section. Multiplication of Polnomials 1

15 Math 1 :: Elementar Algebra 9. x x x x x 8x 11. p p p 0. m m m 1. x 1 8x 1 x. x x x. a ba ab b 7. x x x 8. Multipl each pair of binomials, and then answer the last question. x x a) b) c) 1 1 p p d) e) a1a 1 f) In our own words, describe how the above problems similar before the are multiplied, how the similar after the are multiplied, and then describe the pattern. 9. Multipl each pair of binomials, and then answer the last question. a) x 1 b) c) 1 k d) a 7 e) x 1 f) In our own words, describe how the above problems similar before the are multiplied, how the similar after the are multiplied, and then describe the pattern. 0. Multipl each pair of binomials, and then answer the last question. a) b) x c) 1 a d) a 1 e) x f) In our own words, describe how the above problems similar before the are multiplied, how the similar after the are multiplied, and then describe the pattern. Section. Multiplication of Polnomials 1

16 Math 1 :: Elementar Algebra Section. Division of Polnomials Examples: Perform each operation. a) 8x x x x To divide a polnomial b a monomial, divide the monomial into each term of the polnomial. Notice that after the division/simplification, there will be the same number of terms in the answer as there were in the polnomial. 8x x x x 8x x x x x x 1 x x1 b) x x 8 x To divide a polnomial b a binomial, use polnomial long division. There is another wa to divide polnomials b binomials of degree 1; this method will be covered in the next math course. x x 8 x x The first step is to figure out what times the first term, x, x x x 8 of the divisor x will be x, the first term of the dividend x x x 8 x x x 7x x x 8. For this problem that value is x, and it is written on the top of the long division bar. Next, multipl that value b the binomial x, and write it below the dividend inside the long division bar, so that like-terms are lined-up. Subtract that product from the polnomial x x 8. x x x 8 x x 7x 8 x 7 7x 1 Now, figure out what times the first term, x, of x will be 7x, the first term of result of the subtraction above, 7x 8. For this problem that value is 7, and it is the next term written on the top of the long division bar. is the remainder. In the answer, it is written over the divisor x. The answer to x x 8 x is x 7 x. You ma alwas check division problems b multipling the divisor b the quotient and adding the remainder. Doing this should result in the dividend. Check: x x 7 x x1 x x 8 Homework 1. In our own words, describe the easiest wa to divide a polnomial b a monomial.. When ou divide a polnomial with n terms b a monomial, how man terms will ou have in our quotient (answer)?. In our own words, describe how to divide a polnomial b a binomial. Section. Division of Polnomials 1

17 Math 1 :: Elementar Algebra Perform each operation. p n 7. 8x k x x x x p 18p 1 p p 1. 9x x 1x x 1. 1k 0k k k m 10 1m 9 m m m 17. x 1 0x 10 18x 1x x p 1p 7 p p 7 p 1 Perform each operation k 8k1 k. x x 1 x. m 10m 7 m. a a 11 a 1. p p1 p 7 Section. Division of Polnomials 1

18 Math 1 :: Elementar Algebra. x 1x 0 x k k 1 k 9. x 8x 10 x m m 1 m 1. k k8 k. p p 11p 10 p x 8 x. x x. x 7 x 7. x 8 x Section. Division of Polnomials 17