Name: Chapter 7: Exponents and Polynomials


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1 Name: Chapter 7: Exponents and Polynomials 71: Integer Exponents Objectives: Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. You have seen positive exponents. Recall that to simplify, use as a factor times: = = 9. But what does it mean for an exponent to be negative or 0? You can use a table and look for a pattern to figure it out. Power Value Example 1: One cup is gallons. Simplify this expression. Zero and Negative Exponents Simplify the following expressions: Example A:  Example a: 10  B: 7 0 b: ()  C: (5)  c: () 5 D: 5  d:  5 Ch. 7 Pg 1
2 Evaluating Expressions with Zero and Negative Exponents Evaluate the expression for the given value of the variables. Example A: x for x = a: p  for p = B: a 0 b  for a = 5 and b =  b: 8a  b 0 for a =  and b = 6 Simplifying Expressions with Zero and Negative Numbers Simplify. Example A: 7w  a: r 0 m  5 k B: b: r 7 a c 0 b d C: 6 g c: 6 h Homework: 71 (pg 9):, 1, 0, 5, 6, 9, 0,,, 7, 9,,, 50, 51, 96, 97, 10, 106, 111 Ch. 7 Pg
3 7: Multiplication Properties of Exponents Objective: Use multiplication properties of exponents to evaluate and simplify expressions. Products of powers with the same base can be found by writing each power as a repeated multiplication. Notice the relationship between the exponents in the factors and the exponents in the product 5 + = 7. Finding Products of Powers Simplify. Example 1A: 5 1a: B: 1b: C: q r 6 q 1c: m n m 1D: n n n 1d: 1 x x x x Ch. 7 Pg
4 Example : Skip To find a power of a power, you can use the meaning of exponents. Notice the relationship between the exponents in the original power and the exponent in the final power: = 6. Finding Powers of Powers Simplify: Example A: ( 5 ) a: ( ) 5 B: ( ) 0 0 b: ( 6 ) x 5 x C: ( ) c: ( a ) ( a ) Powers of products can be found by using the meaning of an exponent. Ch. 7 Pg
5 Finding Powers of Products Simplify: Example A: ( y) a: ( p ) B: ( y) b: ( 5t ) 6 C: ( x y ) c: ( x y ) ( x y ) Homework: 7 (Pg 6), 10, 1, 18 1,,, 6, 7, 9, 0,,, 50, 550, 80 Ch. 7 Pg 5
6 7: Division Properties of Exponents Objective: Use division properties of exponents to evaluate and simplify expressions. A quotient of powers with the same base can be found by writing the powers in a factored form and dividing out common factors. Notice the relationship between the exponents in the original quotient and the exponent in the final answer: 5 =. Finding Quotients of Powers Simplify. 7 Example 1A: 1a: 7 9 x x 1B: y 1b: y d 1C: ( ) de e m 5 n 1c: 5 ( m ) n Example : Skip Example : Skip Ch. 7 Pg 6
7 A power of a quotient can be found by first writing the numerator and denominator as powers. Notice that the exponents in the final answer are the same as the exponent in the original expression. Finding Positive Powers of Quotient Simplify Example A: 7 a: B: d ef b: ab c d 5 Ch. 7 Pg 7
8 Ch. 7 Pg 8 Finding Negative Powers of Quotients Simplify. Example 5A: 5B: y x 5a: 5b: c b a Homework: 7 (Pg 71):, 10, 1, 17, 19, 0,, 5, 8, 9, 56, 57
9 76: Adding and Subtracting Polynomials Objective: Add and subtract polynomials. Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms. Adding and Subtracting Monomials Add or Subtract. Example 1A: 1p + 11p + 8p 1a: s + s + s 1B: t + s t s 1b: z z + 1C: 10m n + m n 8m n 1c: 9b c + 5b c 1b c Adding Polynomials Add. Example A: (m + 5) + (m m + 6) C: (6x y) + (x + y 8x y) B: (10xy + x) + ( xy + y) a: (5a + a 6a + 1a ) + (7a 10a) Ch. 7 Pg 9
10 To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial: (x x + 7)= x + x 7 Subtracting Polynomials Subtract: Example A: (x + y) (x ) C: ( 10x x + 7) (x 9) B: (7m m ) (5m 5m + 8) a: (x x + 1) (x + x + 1) Homework: 76 (Pg 87): 1, 7, 1, 16, 17, 19, 0,,, 5, 5, 55 Ch. 7 Pg 10
11 77: Multiplying Polynomials Objective: Multiply polynomials. To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter. Multiplying Monomials Multiply: Example 1A: (6y )(y 5 ) 1a: (x )(6x ) 1B: (mn ) (9m n) 1b: (r t)(5t ) To multiply a polynomial by a monomial, use the Distributive Property. Multiplying a Polynomial by a Monomial Multiply: Example A: (x + x 8) a: (x + x + ) B: 6pq(p q) b: ab(5a + b) To multiply a binomial by a binomial, you can apply the Distributive Property more than once: A method for multiplying binomials is called the FOIL method. (x + )(x + ) 1. Multiply the First terms. (x + )(x + ). Multiply the Outer terms. (x + )(x + ). Multiply the Inner terms. (x + )(x + ). Multiply the Last terms. (x + )(x + ) Ch. 7 Pg 11
12 Multiplying Binomials Multiply: Example A: (s + )(s ) a: (a + )(a ) B: (x ) b: (x ) C: (8m n)(m n) c: (a b )(a + b ) To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + ) by (x + 10x 6): (5x + )(x + 10x 6) = 5x(x + 10x 6) + (x + 10x 6) Multiplying Polynomials Multiply: Example A: (x 5)(x + x 6) a: (x + )(x x + 6) B: (x 5)( x 10x + ) b: (x + )(x x + 5) Homework: 77 (Pg 97): 1, 10, 1, 19, 6, 7, 9, 0,,, 5, 6, 8, 9, 1,,, 5, 7, 8, 50, 51, 5, 56, 88 Ch. 7 Pg 1
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