1. Factors, Remainders, and Roots, Oh My! 1 November 2010

2. Remainders Is there any way I can figure out my remainder in advance? (3x 4 – 8x 3 + 9x + 5) ÷ (x – 2) 3x 3 – 2x 2 – 4x + 1 Remainder 7

3. Remainder Theorem If a polynomial f(x) is divided by x – c, then the remainder is f(c). Like synthetic division, the divisor must be in the form x – c. If it isn’t, we must alter to the divisor to include subtraction.

4. Remainder Theorem, cont. (3x 4 – 8x 3 + 9x + 5) ÷ (x – 2) f(2) = 3(2) 4 – 8(2) 3 + 9(2) + 5 f(2) = 7

5. Remainder Theorem, cont. (2x 4 + 5x 3 − 2x − 8) ÷ (x + 3) x + 3 x – (-3) f(-3) = 2(-3) 4 + 5(-3) 3 – 2(-3) – 8 f(-3) = 25

6. Your Turn On page 249 in your textbook, complete problems 10 – 16. You will Solve for the quotient using synthetic division Check your remainder using the Remainder Theorem

7. Remainders and Factors If a polynomial f(x) is divided by x – a, and f(a) = 0, then x – a is a factor of the polynomial. The Factor Theorem

8. Remainders and Factors, cont. Similarly, if a divisor has a remainder of zero, than the quotient is also a factor of the polynomial.

9. Remainders and Factors, cont. Ex. (a 4 – 1) ÷ (a – 1) = a 3 + a 2 + a + 1 Both a – 1 and a 3 + a 2 + a + 1 are factors of a 4 – 1!

10. Your Turn: On pg. 249 in your textbook, complete problems 41 – 46. You will: Use the Factor Theorem to determine if the given h(x) is a factor of f(x). Confirm your results using synthetic division.

11. Maximum Number of Roots A polynomial of degree n has at most n different roots. Example: f(x) = x 2 – 3x + 4 has at most 2 different roots 0 = (x – 3)(x – 1); x = 1, 3

12. Maximum Number of Roots, cont. However, a polynomial can have less than the maximum number of different roots. This is because roots can repeat. Example: f(x) = x 2 – 10x + 25 0 = (x – 5)(x – 5); x = 5

13. Other Roots Connections Let f(x) be a polynomial. If r is a real number for which one of the following statements is true, then all of the following statements are true: r is a zero of f(x)

14. Other Roots Connections, cont. r is an x-intercept of f(x) x = r is a solution or root when f(x) = 0 x – r is a factor of the polynomial f(x)

15. Applications We can use the maximum number of roots and the root connections to construct the equation of a polynomial from its graph.

16. Applications, cont. x-intercepts: Zeros: Solutions: Max Degree: Linear Factors:

17. Applications, cont. Linear Factors: (x+1)(x – 3) Equation:

18. Your Turn: On page 249 in your textbook, complete problems 51 – 53. You will: List the x-intercepts List the zeros List the solutions Determine the maximum degree Product of the linear factors Determine the equation of a graph

19. Hmwk: Pg. 317: 1 – 5