1 / 14

Warm up!!

Warm up!!. Pg. 130 #’s 16, 22, 36, 42, 52, 96. Divide Factor Find Zeros Example: Divide 6x 3 – 19x 2 + 16x – 4 by x – 2 Answer: 6x 2 – 7x + 2 Which means that… 6x 3 – 19x 2 + 16x – 4 = (x – 2)(6x 2 – 7x + 2) 6x 3 – 19x 2 + 16x – 4 = (x – 2)(2x – 1)(3x – 2)

madeline-willis
Download Presentation

Warm up!!

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. Warm up!! • Pg. 130 #’s 16, 22, 36, 42, 52, 96

  2. Divide • Factor • Find Zeros Example: Divide 6x3 – 19x2 + 16x – 4 by x – 2 Answer: 6x2 – 7x + 2 Which means that… 6x3 – 19x2 + 16x – 4 = (x – 2)(6x2 – 7x + 2) 6x3 – 19x2 + 16x – 4 = (x – 2)(2x – 1)(3x – 2) Zeros: x = 2 x = ½ x = ⅔ Answer: 5x + 3 Can check zeros on calculator

  3. Rules of Long Division • If the divisor goes into the equation evenly then it is a factor and a zero. • If the quotient has a remainder then it is not a factor. • The proper form of writing the remainder is adding it to the quotient and over the divisor.

  4. Also written as: F(x) = q(x) + r(x) d(x) d(x) Improper Proper

  5. Using Division Algorithm you can write it like this: x3 – 1 = x2 + x + 1 x – 1 Example: Divide x3-1 by x-1 • Write in descending powers • Insert zeros where there are missing power x2 + x + 1 x – 1 √(x3 + 0x2 + 0x – 1) x3 – x2 x2 + 0x x2 – x x – 1 x – 1 0 Because it equals 0, x3 -1 is divisible by x-1

  6. Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor. • Used when finding zeroes of polynomials. Example: Divide x2 + 5x + 6 by x – 1 *Rational roots test determines that +/- 1, 2, 3, and 6 are possible zeros. • Synthetic provides the same quotient but in a quicker fashion.

  7. Let’s divide:x2 + 5x + 6 by x – 1

  8. If r = 0, (x – k) is a factor. • If r = 0, (k, 0) is an x intercept of f. *k is a zero • The remainder r gives the value of f at x = k, if r = f(k) (Remainder Theorem)

  9. To evaluate polynomial function f(x) when x=k, divide f(x) by x-k • Remainder will equal f(k) Example 1: Evaluate f(x) = 3x3 + 8x2 + 5x – 7 for x = -2 So f(-2) = -9, r = -9 Example 2: Evaluate f(x) = x3 – x2 – 14x + 11 for x = 4 So f(4) = 3, r = 3

  10. Repeated Division • Multi-step synthetic division… Find all the zeros of the following polynomial function given (x – 2) and (x + 3) are factors… f (x) = 2x4 + 7x3 – 4x2– 27x – 18. Solution:Using synthetic division with the factor (x – 2), you obtain the following. 0 remainder, so f(2) = 0 and (x – 2) is a factor.

  11. Repeated Division (Continued) Take the result of this division and perform synthetic division again using the factor (x + 3). Because the resulting quadratic expression factors as 2x2 + 5x + 3 = (2x + 3)(x + 1) the complete factorization of f (x) is f (x) = (x – 2)(x + 3)(2x + 3)(x + 1). 0 remainder, so f(-3) = 0 and (x +3 ) is a factor. Try #60 on page 141

  12. Example 1: Show that (x – 2) and (x + 3) are factors and find all the zeros of f(x) = 2x4 + 7x3 – 4x2 – 27x – 18 x = 2, -3, -3/2, -1 Example 2: Show that (x + 2) and (x – 1) are factors and find all the zeros of f(x) = 2x3 + x2 – 5x + 2 x = ½, -2, 1 • Graph should confirm zeros

  13. Page 140 #’s 5 – 35 (5’s), 47, 51, 55, 59, 61, 69, 73

More Related