Polynomials, Linear Factors, Zeros

1. Polynomials, Linear Factors, Zeros 1 Definitions 2 Relative Minimum/Maximum 3 Factors & Zeros

2. Definitions • Factor • Each term in the completely factored form of a polynomial • Can have repeated factors • Zeros • Values found when each factor is set equal to zero and solved • Also called the x-intercepts or roots • Multiplicity of zeros • The number of times a root is repeated • i.e. (x-2)2 = (x-2)(x-2), so the solution is x=2 and x=2

3. Definitions • Relative minimum • The value of a function at a down-to-up turning point • Relative Maximum • The value of a function at a up-to-down turning point

4. Relative Minimum/Maximum Relative Maximum Relative Minimum

5. Relative Minimum/Maximum Degree = 3 3-1=2

6. Finding the Zeros from a Polynomial • Write the polynomial in standard form • Set the polynomial equal to zero • Factor using GCF • Factor using “Rainbow Method” or Special Rules • Set each factor equal to zero • Solve each factor

7. GCF Review • Look for the largest common number and/or variables in each term • Divide each term by the common terms

8. Rainbow Method Review

9. Factoring Example • Factor

10. Finding the Zeros Example • Find the zeros for

11. Writing a Polynomial from the Zeros • Substitute each zero (root) for a in (x-a) • Multiply the (x-a)’s together • Combine like terms

12. Writing a Polynomial from Zeros Example • What polynomial function in standard form has zeros -2, 2, and 3?