1. Find the roots Identify the multiplicity 3.5: Finding Real Roots of Polynomial Equations

2. Common Core Standards: CC.9-12.A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. CC.9-12.A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, … Include cases where f(x) and/or g(x) are linear, polynomial, rational, … functions. 3.5: Finding Real Roots of Polynomial Equations

3. Solving a Polynomial: P(x) = 0 If (x-a) is a factor of P(x), then x=a is a solution. If x=a is a solution of P(x), then (x-a) is a factor. (Factor Theorem)

4. Example

5. Steps to Solve a Polynomial Equation: 1.Factor out any GCF 2.Graph and identify a root. 3.Use synthetic division to prove it is a root and simplify the equation down to a quadratic. 4.Finish solving by using the quadratic formula (or factoring). 5.CHECK your answers (look at the graph!)

6. Solve the polynomial equation by factoring. Example 1: Using Factoring to Solve Polynomial Equations 4x 6 + 4x 5 – 24x 4 = 0 Factor out the GCF, 4x 4. Factor the quadratic or use Quadratic Formula. Set each factor equal to 0. Solve for x. The roots are 0, –3, and 2.

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

8. Solve the polynomial equation by factoring. Example 1: Using Factoring to Solve Polynomial Equations x 4 + 25 = 26x 2 Set each factor equal to 0. Solve for x. The roots are 5, –5, 1, and -1. Set the equation equal to 0. Factor the trinomial. Factor - difference of squares.

9. Multiplicity

10. 3x 5 + 18x 4 + 27x 3 = 0 Factored form:3x 3 (x + 3) 2 =0 x is a factor three times. Therefore, the multiplicity of the root (0) is… (x+3) is a factor two times. Therefore, the multiplicity of the root (-3) is… Multiplicity

11. You cannot always determine the multiplicity of a root from a graph. You determine multiplicity when the polynomial is in factored form. 3x 5 + 18x 4 + 27x 3 = 0 3x 3 (x + 3) 2 =0

12. Identify the roots of each equation. State the multiplicity of any root. 4x 6 + 4x 5 – 24x 4 = 0 4x 4 (x + 3)(x – 2) = 0 The roots are 0, –3, and 2. Zero has a multiplicity of 4. 4x 4 (x 2 + x – 6) = 0

13. Identify the roots of each equation. State the multiplicity of any root. x 3 + 3x 2 - 10x = 0 x (x + 5)(x – 2) = 0 The roots are 0, –5, and 2. No Multiplicities in these zeros.

14. Identify the roots of each equation. State the multiplicity of any root. (x – 1)(x + 3)(x + 3)(x + 3)=0 The roots are 1 and -3. The root –3 has a multiplicity of 3. x 4 + 8x 3 + 18x 2 – 27 = 0

15. Solving a Polynomial: P(x) = 0 If (x-a) is a factor of P(x), then x=a is a solution. If x=a is a solution of P(x), then (x-a) is a factor. (Factor Theorem)

16. Given the roots of a polynomial, find the polynomial:

18. Homework 3.5, p 186: # 15-22 WS - Odds

19. HA2 Homework 3.5, pp 185-187: # 15-26, 44