1 / 17

# 6-5

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

Télécharger la présentation

## 6-5

An Image/Link below is provided (as is) to download presentation Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

### Presentation Transcript

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.

4. Vocabulary multiplicity

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.

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 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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). 

13. 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). 

14. 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). 

15. 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.

16. 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

More Related