7.5 Solving Polnomial Equations Eponential Growth in Factored Form is written in factored form? How can ou solve a polnomial equation that Two polnomial equations are equivalent when the have the same solutions. For instance, the following equations are equivalent because the onl solutions of each equation are = and =. Factored Form ( )( ) = 0 Standard Form + = 0 Nonstandard Form = Check this b substituting and for in each equation. ACTIVITY: Matching Equivalent Forms of an Equation Work with a partner. Match each factored form of the equation with two other forms of equivalent equations. Notice that an equation is considered to be in factored form onl when the product of the factors is equal to 0. Factored Form Standard Form a. ( )( ) = 0 A. = 0. b. ( )( ) = 0 B. + = 0. c. ( + )( ) = 0 C. 4 + = 0. d. ( )( + ) = 0 D. 5 + 6 = 0 4. e. ( + )( ) = 0 E. = 0 5. Nonstandard Form 5 = 6 ( ) = 4 = ( + ) = 4 = COMMON CORE Polnomial Equations In this lesson, ou will solve polnomial equations in factored form. Learning Standard A.REI.4b ACTIVITY: Writing a Conjecture Work with a partner. Substitute,,, 4, 5, and 6 for in each equation. Write a conjecture describing what ou discovered. a. ( )( ) = 0 b. ( )( ) = 0 c. ( )( 4) = 0 d. ( 4)( 5) = 0 e. ( 5)( 6) = 0 f. ( 6)( ) = 0 56 Chapter 7 Polnomial Equations and Factoring
ACTIVITY: Special Properties of 0 and Work with a partner. The numbers 0 and have special properties that are shared b no other numbers. For each of the following, decide whether the propert is true for 0,, both, or neither. Eplain our reasoning. a. If ou add to a number n, ou get n. b. If the product of two numbers is, then one or both numbers are 0. c. The square of is equal to itself. d. If ou multipl a number n b, ou get n. e. If ou multipl a number n b, ou get 0. f. The opposite of is equal to itself. 4 ACTIVITY: Writing About Solving Equations Math Practice Use Definitions What previous eamples, information, and definitions can ou use to repl to the student s comment? Work with a partner. Imagine that ou are part of a stud group in our algebra class. One of the students in the group makes the following comment. I don t see wh we spend so much time solving equations that are equal to zero. Wh don t we spend more time solving equations that are equal to other numbers? Write an answer for this student. 5. One of the properties in Activit is called the Zero-Product Propert. It is one of the most important properties in all of algebra. Which propert is it? Eplain how it is used in algebra and wh it is so important. 6. IN YOUR OWN WORDS How can ou solve a polnomial equation that is written in factored form? Use what ou learned about solving polnomial equations to complete Eercises 4 6 on page 60. Section 7.5 Solving Polnomial Equations in Factored Form 57
7.5 Lesson Lesson Tutorials Ke Vocabular factored form, p. 58 Zero-Product Propert, p. 58 root, p. 58 A polnomial is in factored form when it is written as a product of factors. Standard form Factored form + ( + ) + 5 4 ( )( + 8) When one side of an equation is a polnomial in factored form and the other side is 0, use the Zero-Product Propert to solve the polnomial equation. The solutions of a polnomial equation are also called roots. Zero-Product Propert Words If the product of two real numbers is 0, then at least one of the numbers is 0. Algebra If a and b are real numbers and ab = 0, then a = 0 or b = 0. EXAMPLE Solving Polnomial Equations Solve each equation. Check Substitute each solution in the original equation. 0(0 + 8) =? 0 0(8) =? 0 0 = 0 8( 8 + 8) =? 0 8(0) =? 0 0 = 0 a. ( + 8) = 0 ( + 8) = 0 Write equation. = 0 or + 8 = 0 Use Zero-Product Propert. = 8 Solve for. The roots are = 0 and = 8. b. ( + 6)( 5) = 0 ( + 6)( 5) = 0 Write equation. + 6 = 0 or 5 = 0 Use Zero-Product Propert. = 6 or = 5 Solve for. The roots are = 6 and = 5. Eercises 4 9. ( ) = 0. t(t + ) = 0. (z 4)(z 6) = 0 4. (b + 7) = 0 58 Chapter 7 Polnomial Equations and Factoring
EXAMPLE Solving a Polnomial Equation What are the solutions of (a + 7)(a 7) = 0? A 7 and 7 B 7 and 7 C and D 7 and 7 (a + 7)(a 7) = 0 Write equation. a + 7 = 0 or a 7 = 0 Use Zero-Product Propert. a = 7 or a = 7 Solve for a. The correct answer is B. EXAMPLE Real-Life Application The arch of a fireplace can be modeled b = ( + 8)( 8), 9 where and are measured in inches. The -ais represents the floor. Find the width of the arch at floor level. Use the -coordinates at floor level to find the width. At floor level, = 0. So, substitute 0 for and solve for. 5 = ( + 8)( 8) Write equation. 9 O 5 0 = ( + 8)( 8) Substitute 0 for. 9 0 = ( + 8)( 8) Multipl each side b 9. + 8 = 0 or 8 = 0 Use Zero-Product Propert. = 8 or = 8 Solve for. The width is the distance between the -coordinates, 8 and 8. So, the width of the arch at floor level is 8 ( 8) = 6 inches. Eercises 0 5 5. (p + 5)(p 5) = 0 6. ( 6) = 0 7. The entrance to a mine shaft can be modeled b = ( + 4)( 4), where and are measured in feet. The -ais represents the ground. Find the width of the entrance at ground level. Section 7.5 Solving Polnomial Equations in Factored Form 59
7.5 Eercises Help with Homework. REASONING Is = a solution of ( )( + 6) = 0? Eplain.. WRITING Describe how to solve ( )( + ) = 0 using the Zero-Product Propert.. WHICH ONE DOESN T BELONG? Which statement does not belong with the other three? Eplain our reasoning. (n 9)(n + ) (k + 5)(k ) (g + ) + 4 9+(-6)= +(-)= 4+(-9)= 9+(-)= 4. ( + 7) = 0 5. t(t 5) = 0 6. (s 9)(s ) = 0 7. (q + )(q ) = 0 8. (h 8) = 0 9. (m + 4) = 0 0. (5 k)(5 + k) = 0. ( g)(7 g) = 0. (p + 6) = 0. (4z ) = 0 4. ( + 4 ) ( 8) = 0 5. ( d ) ( d + ) = 0 6. ERROR ANALYSIS Describe and correct the error in solving the equation. 6( + 5) = 0 + 5 = 0 = 5 The root is = 5. Find the -coordinates of the points where the graph crosses the -ais. 7. = ( 8)( + 8) 8. = ( 4)( 5) 9. = 0.( + )( 5) 5 9 O 6 9 0 60 6 5 50 4 0 5 40 0 40 0 O 8 6 0 0 0 5 0 O 0 0 0 60 Chapter 7 Polnomial Equations and Factoring
0. CHOOSE TOOLS The entrance of a tunnel can be modeled b = ( 4)( 4), where 50 and are measured in feet. The -ais represents the ground. Find the width of the tunnel at ground level. 4 O 8. 5z(z + )(z ) = 0. w(w 6) = 0. (r 4)(r + 4)(r + 8) = 0 4. (p + )(p )(p + 7) = 0 00 O Tallest point 00 5. GATEWAY ARCH The Gatewa Arch in St. Louis can be modeled b = ( + 5)( 5), 5 where and are measured in feet. The -ais represents the ground. a. Find the width of the arch at ground level. b. How tall is the arch? 6. Find the values of in terms of that are solutions of the equation. a. ( + )( ) = 0 b. ( )(4 + 6) = 0 Find the greatest common factor of the numbers. (Skills Review Handbook) 7. and 6 8. and 7 9. 0, 75, and 90 0. MULTIPLE CHOICE What is the slope of the line? (Section.) A B 4 4 C D 4 Section 7.5 Solving Polnomial Equations in Factored Form 6