1 / 19

Factors, Remainders, and Roots, Oh My!

Factors, Remainders, and Roots, Oh My!. 1 November 2010. 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. Remainder Theorem. If a polynomial f(x) is divided by x – c , then the remainder is f(c).

amaris
Download Presentation

Factors, Remainders, and Roots, Oh My!

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Remainders • Is there any way I can figure out my remainder in advance? • (3x4 – 8x3 + 9x + 5) ÷ (x – 2) • 3x3 – 2x2 – 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. • (3x4 – 8x3 + 9x + 5) ÷ (x – 2) • f(2) = 3(2)4 – 8(2)3 + 9(2) + 5 • f(2) = 7

  5. Remainder Theorem, cont. • (2x4 + 5x3 − 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. (a4 – 1) ÷ (a – 1) = a3 + a2 + a + 1 • Both a – 1 and a3 + a2 + a + 1 are factors of a4 – 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) = x2 – 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) = x2 – 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

More Related