Finding Real Roots of of Finding Real Roots
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Finding Real Roots of of Finding Real Roots 6 -5 Polynomial Equations Warm Up Lesson Presentation Lesson Quiz Holt Algebra 22
6 -5 Finding Real Roots of Polynomial Equations Warm Up Factor completely. 1. 2 y 3 + 4 y 2 – 30 y 2 y(y – 3)(y + 5) 2. 3 x 4 – 6 x 2 – 24 3(x – 2)(x + 2)(x 2 + 2) Solve each equation. 3. x 2 – 9 = 0 x = – 3, 3 4. x 3 + 3 x 2 – 4 x = 0 Holt Algebra 2 x = – 4, 0, 1
6 -5 Finding Real Roots of Polynomial Equations Objectives Identify the multiplicity of roots. Factor polynomial expressions and find the roots. Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Vocabulary multiplicity The number of times that x-r is a factor of P(x) When a real root has even multiplicity, the graph of P(x) touches the x-axis but does not cross it. When a real root has odd multiplicity greater than 1, the graph “bends” as it crosses the x-axis. Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations In Lesson 6 -4, you used several methods for factoring polynomials. As with some quadratic equations, factoring a polynomial equation is one way to find its real roots. Recall the Zero Product Property from Lesson 5 -3. You can find the roots, or solutions, of the polynomial equation P(x) = 0 by setting each factor equal to 0 and solving for x. ROXS Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Example 1 A: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. 4 x 6 + 4 x 5 – 24 x 4 = 0 4 x 4(x 2 + x – 6) = 0 Factor out the GCF, 4 x 4(x + 3)(x – 2) = 0 Factor the quadratic. 4 x 4 = 0 or (x + 3) = 0 or (x – 2) = 0 Set each factor equal to 0. Solve for x. x = 0, x = – 3, x = 2 The roots are 0, – 3, and 2. Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Example 1 A Continued Check Use a graph. The roots appear to be located at x = 0, x = – 3, and x = 2. Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Example 1 B: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. x 4 + 25 = 26 x 2 x 4 – 26 x 2 + 25 = 0 (x 2 – 25)(x 2 – 1) = 0 Set the equation equal to 0. Factor the trinomial in quadratic form. (x – 5)(x + 5)(x – 1)(x + 1) Factor the difference of two squares. x – 5 = 0, x + 5 = 0, x – 1 = 0, or x + 1 =0 x = 5, x = – 5, x = 1 or x = – 1 The roots are 5, – 5, 1, and – 1. Holt Algebra 2 Solve for x.
6 -5 Finding Real Roots of Polynomial Equations Check It Out! Example 1 a Solve the polynomial equation by factoring. 2 x 6 – 10 x 5 – 12 x 4 = 0 2 x 4(x 2 – 5 x – 6) = 0 Factor out the GCF, 2 x 4(x – 6)(x + 1) = 0 Factor the quadratic. 2 x 4 = 0 or (x – 6) = 0 or (x + 1) = 0 Set each factor equal to 0. x = 0, x = 6, x = – 1 Solve for x. The roots are 0, 6, and – 1. Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Check It Out! Example 1 b Solve the polynomial equation by factoring. x 3 – 2 x 2 – 25 x = – 50 x 3 – 2 x 2 – 25 x + 50 = 0 Set the equation equal to 0. (x + 5)(x – 2)(x – 5) = 0 Factor. x + 5 = 0, x – 2 = 0, or x – 5 = 0 x = – 5, x = 2, or x = 5 Solve for x. The roots are – 5, 2, and 5. Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root. In 3 x 5 + 18 x 4 + 27 x 3 = 0 has two multiple roots, 0 and – 3. For example, the root 0 is a factor three times because 3 x 3 = 0. The multiplicity of root r is the number of times that x – r is a factor of P(x). When a real root has even multiplicity, the graph of y = P(x) touches the x-axis but does not cross it. When a real root has odd multiplicity greater than 1, the graph “bends” as it crosses the x-axis. Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations You cannot always determine the multiplicity of a root from a graph. It is easiest to determine multiplicity when the polynomial is in factored form. Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Example 2 A: Identifying Multiplicity Identify the roots of each equation. State the multiplicity of each root. x 3 + 6 x 2 + 12 x + 8 = 0 x 3 + 6 x 2 + 12 x + 8 = (x + 2)(x + 2) x + 2 is a factor three times. The root – 2 has a multiplicity of 3. Check Use a graph. A calculator graph shows a bend near (– 2, 0). Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Example 2 B: Identifying Multiplicity Identify the roots of each equation. State the multiplicity of each root. x 4 + 8 x 3 + 18 x 2 – 27 = 0 x 4 + 8 x 3 + 18 x 2 – 27 = (x – 1)(x + 3)(x + 3) x – 1 is a factor once, and x + 3 is a factor three times. The root 1 has a multiplicity of 1. The root – 3 has a multiplicity of 3. Check Use a graph. A calculator graph shows a bend near (– 3, 0) and crosses at (1, 0). Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Check It Out! Example 2 a Identify the roots of each equation. State the multiplicity of each root. x 4 – 8 x 3 + 24 x 2 – 32 x + 16 = 0 x 4 – 8 x 3 + 24 x 2 – 32 x + 16 = (x – 2)(x – 2) x – 2 is a factor four times. The root 2 has a multiplicity of 4. Check Use a graph. A calculator graph shows a bend near (2, 0). Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Check It Out! Example 2 b Identify the roots of each equation. State the multiplicity of each root. 2 x 6 – 22 x 5 + 48 x 4 + 72 x 3 = 0 2 x 6 – 22 x 5 + 48 x 4 + 72 x 3 = 2 x 3(x + 1)(x – 6) x is a factor three times, x + 1 is a factor once, and x – 6 is a factor two times. The root 0 has a multiplicity of 3. The root – 1 has a multiplicity of 1. The root 6 has a multiplicity of 2. Holt Algebra 2
6 -5 Finding Real Roots of Polynomial Equations Lesson Quiz Solve by factoring. 1. x 3 + 9 = x 2 + 9 x – 3, 3, 1 Identify the roots of each equation. State the multiplicity of each root. 0 and 2 each with 2. 5 x 4 – 20 x 3 + 20 x 2 = 0 multiplicity 2 3. x 3 – 12 x 2 + 48 x – 64 = 0 Holt Algebra 2 4 with multiplicity 3
6 -5 Finding Real Roots of Polynomial Equations SUGGESTED PRACTICE PAGE 442 #1 -9, 15 -22 AND 43 Holt Algebra 2