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  1. Finding Real Roots of Polynomial Equations 6-5 Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

  2. Warm Up Factor completely. 1. 2y3 + 4y2 – 30y 2y(y – 3)(y + 5) 2. 3x4 – 6x2 – 24 3(x – 2)(x + 2)(x2 + 2) Solve each equation. 3. x2 – 9 = 0 x = – 3, 3 4. x3 + 3x2 – 4x = 0 x = –4, 0, 1

  3. Objectives Identify the multiplicity of roots. Factor polynomial expressions and find the roots.

  4. 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.

  5. 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

  6. Example 1A: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. 4x6 + 4x5 – 24x4 = 0 4x4(x2 + x – 6) = 0 Factor out the GCF, 4x4. 4x4(x+ 3)(x – 2) = 0 Factor the quadratic. 4x4 = 0 or (x+ 3) = 0 or (x – 2) = 0 Set each factor equal to 0. x = 0, x = –3, x = 2 Solve for x. The roots are 0, –3, and 2.

  7. Example 1A Continued Check Use a graph. The roots appear to be located at x = 0, x = –3, and x = 2. 

  8. Example 1B: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. x4 + 25 = 26x2 x4 – 26 x2 + 25 = 0 Set the equation equal to 0. Factor the trinomial in quadratic form. (x2 – 25)(x2 – 1) = 0 (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 Solve for x. The roots are 5, –5, 1, and –1.

  9. Check It Out! Example 1a Solve the polynomial equation by factoring. 2x6 – 10x5 – 12x4 = 0 2x4(x2 – 5x – 6) = 0 Factor out the GCF, 2x4. 2x4(x– 6)(x + 1) = 0 Factor the quadratic. 2x4 = 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.

  10. Check It Out! Example 1b Solve the polynomial equation by factoring. x3 – 2x2 – 25x = –50 x3 – 2x2 – 25x + 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.

  11. Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root. In 3x5 + 18x4 + 27x3 = 0 has two multiple roots, 0 and –3. For example, the root 0 is a factor three times because 3x3 = 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.

  12. 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.

  13. Example 2A: Identifying Multiplicity Identify the roots of each equation. State the multiplicity of each root. x3 + 6x2 + 12x + 8 = 0 x3 + 6x2 + 12x + 8 = (x + 2)(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). 

  14. Example 2B: Identifying Multiplicity Identify the roots of each equation. State the multiplicity of each root. x4 + 8x3 + 18x2 – 27 = 0 x4 + 8x3 + 18x2 – 27= (x – 1)(x + 3)(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). 

  15. Check It Out! Example 2a Identify the roots of each equation. State the multiplicity of each root. x4 – 8x3 + 24x2 – 32x + 16 = 0 x4 – 8x3 + 24x2 – 32x + 16= (x– 2)(x– 2)(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). 

  16. Check It Out! Example 2b Identify the roots of each equation. State the multiplicity of each root. 2x6 – 22x5 + 48x4 + 72x3 = 0 2x6 – 22x5 + 48x4 + 72x3= 2x3(x + 1)(x – 6)(x – 6) xis 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.

  17. Lesson Quiz Solve by factoring. 1. x3 + 9 = x2 + 9x –3, 3, 1 Identify the roots of each equation. State the multiplicity of each root. 0 and 2 each with multiplicity 2 2. 5x4– 20x3 + 20x2 = 0 3. x3 – 12x2+48x – 64 = 0 4 with multiplicity 3

  18. SUGGESTED PRACTICE PAGE 442 #1-9, 15-22 AND 43