Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.

Transcription

1 LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in the Denominator Definition For any nonzero number x, x 0. For any nonzero number x and any integer n, x n n x. For any nonzero number x and any integer n, n x x n. Examples and 0 n are undefined. Simplify. Simplify x y z 0. x y z 0 6 Write without negative exponents. Write in expanded form. x z y 0 x () y x y Write without negative exponents. z 0. Fill in the blanks to simplify each expression y a x b x y a b 9xy x y

2 LESSON 6- Review for Mastery Integer Exponents continued Evaluate a b for a and b. a b ( )( ) Substitute. 6 Write without negative exponents. When evaluating, it is important to determine whether the negative is raised to the power. Evaluate x for x 0. Evaluate ( x) for x 0. The negative is not raised to the power. The negative is raised to the power. x ( x) 0 Substitute. ( 0) Substitute. 0 Write in expanded form Write without negative exponents ( 0) ( 0) ( 0) 00 Write without negative exponents Write in expanded form. Evaluate each expression for the given value(s) of the variable(s). 0. x y 0 for x and y. a b for a and b. z for z and y. a b for a and b y. n m for m 6 and n. ( u) v 6 for u and v

3 6. Review for Mastery Rational Exponents To simplify a number raised to the power of, write the nth root of the number. n Simplify Think: What number, when taken as a factor times, is equal to 6? , so 6 6. When an expression contains two or more expressions with fractional exponents, evaluate the expressions with the exponents first, then add or subtract. Simplify Simplify each expression

4 6. Review for Mastery Rational Exponents continued A fractional exponent may have a numerator other than. To simplify a number raised to the power of m, write the nth root of the number raised to the mth power. n Simplify. 6 To find, think: what number, when taken as a factor times, equals?, so Simplify To find 6, think: what number, when taken as a factor 6 times, equals 6? 6 6 6, so 6. Simplify each expression

5 6. Review for Mastery Polynomials A monomial is a number, a variable, or a product of numbers and variables with wholenumber exponents. A polynomial is a monomial or a sum or difference of monomials. The degree of the monomial is the sum of the exponents in the monomial. Find the degree of 8x y. Find the degree of a 6 b. 8x y The exponents are and. a 6 b The exponents are 6 and. The degree of the monomial The degree of the monomial is. is 6 7. The degree of the polynomial is the degree of the term with the greatest degree. Find the degree of x y 9x. Find the degree of ab 9a. x y 9{ x { ab 9 { a 7 Degree of the polynomial is 7. The standard form of a polynomial is written with the terms in order from the greatest degree to the least degree. The coefficient of the first term is the leading coefficient. Write x 6x x in standard form. { x 6 { x { 0 x 6x x { x The leading coefficient is. Find the degree of each term. Write the terms in order of degree. Degree of the polynomial is. Find the degree of each monomial.. 7m n. 6xyz. x y Find the degree of each polynomial.. x x y. x y y 7 6. x xy y Write each polynomial in standard form. Then give the leading coefficient. 7. x x 6x 8. x x x 9. 8x 7x

6 6. Review for Mastery Polynomials continued Polynomials have special names based on their degree and the number of terms they have. Degree 0 6 or more Name Constant Linear Quadratic Cubic Quartic Quintic 6th degree Name Monomial Binomial Trinomial Polynomial Classify 7x x according to its degree and number of terms. 7x x is a quartic trinomial. Polynomials can be evaluated. A ball is thrown straight up in the air from a height of feet at a speed of 6 feet per second. The height of the ball in feet is given by 6t 6t where t is the time in seconds. How high is the ball after seconds? Evaluate for t. 6t 6t 6() 6() 6() 6() After seconds, the ball is 70 feet high. Terms or more Degree: Terms: Substitute for t. Follow the order of operations to simplify. Classify each polynomial according to its degree and number of terms. 0. 7x x. b b b. A ball is thrown straight up in the air from a height of 6 feet at a speed of 80 feet per second. The height of the ball in feet is given by 6t 80t 6 where t is the time in seconds. What is the height of the ball after seconds?

7 6. Review for Mastery Adding and Subtracting Polynomials You can add or subtract polynomials by combining like terms. The following are like terms: The following are not like terms: Add x x x 6x. x x x 6x x x x 6x 8x 0x Identify like terms. Rearrange terms so that like terms are together. Add (y 7y ) (y y 8). (y 7y ) (y y 8 ) Identify like terms. (y y ) ( 7y y ) ( 8 ) Rearrange terms so that like terms are together. 9y 8y 0 Determine whether the following are like terms. Explain.. x and x. y and 7y. z and x Add.. y y 7y y. 8m m m 6. x 0x 8x 7. (6x x) (x 6x) 8. (m 0m ) (8m ) 9. (6x x) (x x x 9) 0. (y 6y ) (y 8y y )

8 6. Review for Mastery Adding and Subtracting Polynomials continued To subtract polynomials you must remember to add the opposite. Find the opposite of (m m ). (m m ) (m m ) m m Subtract (x x 7) (x ). (x x 7) (x ) Write the opposite of the polynomial. Write the opposite of each term in the polynomial. (x x 7) ( x ) Rewrite subtraction as addition of the opposite. (x x 7) ( x ) Identify like terms. (x x ) x 7 Rearrange terms so that like terms are together. x x 7 Subtract (6y y 7) (y y ). (6y y 7) (y y ) (6y y 7) ( y y ) Rewrite subtraction as addition of the opposite. (6y y 7 ) ( y y ) Identify like terms. (6y y ) ( y y ) ( 7 ) Rearrange terms so that like terms are together. y y Find the opposite of each polynomial.. x 7x. x x 8. x x 7x Subtract.. (9x x) (x). (6t ) ( t ) 6. (x x ) (x 6) 7. (t t) (t t 6) 8. (c 8c c ) (c c )

9 6. Review for Mastery Multiplying Polynomials To multiply monomials, multiply the constants, then multiply variables with the same base. Multiply (a b) (ab ). (a b) (ab ) ( ) (a a) (b b ) Rearrange so that the constants and the variables with the same bases are together. a b To multiply a polynomial by a monomial, distribute the monomial to each term in the polynomial. Multiply x(x + x + 7). x(x x 7) (x)x (x)x (x)7 x 6x x Distribute.. ( x y ) (xy). (xyz) ( x yz). (x) (x y ) Fill in the blanks below. Then finish multiplying.. (x ). x(x 8) 6. x(x 6x ) x x + 8 x 6x 7. (x 9) 8. x(x 8) 9. x (x x ) 0. ( x ). (a b) (ab). y( y 7y )

10 6. Review for Mastery Multiplying Polynomials continued Use the Distributive Property to multiply binomials and polynomials. Multiply (x ) (x 7). (x ) (x 7) x(x 7) (x 7) (x)x (x)7 ()x ()7 x 7x x x x Multiply (x ) (x x ). (x ) (x x ) x(x x ) (x x ) Distribute. Distribute again. (x)x (x)x (x) ()x ()x () x x x x x 0 x 8x 9x 0 Distribute. Distribute again. Fill in the blanks below. Then finish multiplying.. (x ) (x ). (x ) (x 8). (x ) (x 6) (x ) (x ) (x 8) (x 8) (x 6) (x 6) 6. (x ) (x ) 7. (x 7) (x 7) 8. (x ) (x ) Fill in the blanks below. Then finish multiplying. 9. (x ) (x x 8) 0. (x ) (6x x ) (x x 8) (x x 8) (6x x ) (6x x )

11 6.6 Review for Mastery Special Products of Binomials A perfect-square trinomial is a trinomial that is the result of squaring a binomial. (a b) a ab b (a b) a ab b Multiply (x ). Multiply (x ). (x ) a: x (x ) a: x b: b: x (x)() Middle term is added. 6x (x)() Middle term is subtracted. x 8x 6 Square a. Square a. 6x x 9 Square b. Add product of, a, and b. Square b. Subtract product of, a, and b. State whether each product will result in a perfect-square trinomial.. (x ) (x ). (x ) (x ). (x 6) (x 6) Fill in the blanks. Then write the perfect-square trinomial.. (x 7). (x ) 6. (x 0) Square a: Square a: Square a: (a)(b): (a)(b): (a)(b): Square b: Square b: Square b: 7. (x 8) 8. (x ) 9. (7x )

12 6.6 Review for Mastery Special Products of Binomials continued When you multiply certain types of binomials, the middle term will be zero. Multiply (a b) (a b). (a b) (a b) a(a b) b(a b) a ab ab b a b Distribute. This type of special product is called a difference of squares. Multiply (x ) (x ). (a b) (a b) a b Square a. Subtract. Multiply (7 8x) (7 8x). (x ) (x ) a: x (7 8x) (7 8x) a: 7 b: b: 8x (x) () (7) (8x) Square b. x 6 9 6x State whether the products will form a difference of squares or a perfect-square trinomial. 0. (x 0) (x 0). (y 6) (y 6). (z ) (z ) Fill in the blanks. Then write the difference of squares. (a 7) (a 7). ( m) ( m). (x ) (x ) Square a: Square a: Square a: Square b: Square b: Square b: 6. (x 8) (x 8) 7. (0 x) (0 x) 8. (x y) (x y)