1 / 15

2.6 Find Rational Zeros pg. 128

2.6 Find Rational Zeros pg. 128. What is the rational zero theorem? What information does it give you?. The rational zero theorem. …. If f (x)=a n x + +a 1 x+a 0 has integer coefficients, then every rational zero of f has the following form: p factor of constant term a 0

hisa
Télécharger la présentation

2.6 Find Rational Zeros pg. 128

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. 2.6 Find Rational Zerospg. 128 What is the rational zero theorem? What information does it give you?

  2. The rational zero theorem … • If f(x)=anx + +a1x+a0 has integer coefficients, then every rational zero of f has the following form: pfactor of constant term a0 q factor of leading coefficient an n =

  3. Example 1: • Find rational zeros of f(x)=x3+2x2-11x-12 • List possible LC=1 CT=-12 X= ±1/1,± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/1 • Test: 1 2 -11 -12 1 2 -11 -12 X=1 1 3 -8 x=-1 -1 -1 12 1 3 -8 -20 1 1 -12 0 • Since -1 is a zero: (x+1)(x2+x-12)=f(x) Factor: (x+1)(x-3)(x+4)=0 x=-1 x=3 x=-4

  4. List the possible rational zeros of f using the rational zero theorem. a. f (x) = x3 + 2x2 – 11x + 12 Factors of the constant term: ± 1, ± 2, ± 3, ± 4, ± 6, ± 12 Factors of the leading coefficient: ± 1 Simplified list of possible zeros: ± 1, ± 2, ± 3, ± 4, ± 6, ± 12

  5. f (x) = 4x4 – x3 – 3x2 + 9x – 10 Factors of the constant term: + 1, + 2, + 5, + 10 Factors of the leading coefficient:+ 1, + 2, + 4

  6. Extra Example: Find rational zeros of: f(x)=x3-4x2-11x+30 • LC=1 CT=30 x= ±1/1, ± 2/1, ±3/1, ±5/1, ±6/1, ±10/1, ±15/1, ±30/1 • Test: 1 -4 -11 30 1 -4 -11 30 x=1 1 -3 -14 x=-1 -1 5 6 1 -3 -14 16 1 -5 -6 36 X=2 1 -4 -11 30 (x-2)(x2-2x-15)=0 2 -4 -30 (x-2)(x+3)(x-5)=0 1 -2 -15 0 x=2 x=-3 x=5

  7. Find all real zeros of f (x) = x3 – 8x2 +11x + 20. SOLUTION STEP 1

  8. Test x =1: 1 1 – 8 11 20 1 – 7 4 1 – 7 4 24 Test x = –1: –1 1 –8 11 20 –1 9 20 1 – 9 20 0 STEP 2 Test these zeros using synthetic division. 1 is not a zero. –1 is a zero

  9. ANSWER The zeros of fare –1, 4, and 5. Because –1 is a zero of f, you can write f (x) = (x + 1)(x2 – 9x + 20). STEP 3 Factor the trinomial in f (x) and use the factor theorem. f (x) = (x + 1) (x2 – 9x + 20) = (x + 1)(x – 4)(x – 5)

  10. Assignment 2.6 p. 132, 1, 4-22 even, 36-37

  11. Find Zeros -leading coefficient is not 1 • f(x)=10x4-3x3-29x2+5x+12 • List: LC=10 CT=12 x= ± 1/1, ± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/1, ± 3/2, ± 1/5, ± 2/5, ± 3/5, ± 6/5, ± 12/5, ± 1/10, ± 3/10, ± 12/10 • w/ so many –sketch graph on calculator and find reasonable solutions: x= -3/2, -3/5, 4/5, 3/2 Check: 10 -3 -29 5 12 x= -3/2 -15 27 3 -12 10 -18 -2 8 0 Yes it works * (x+3/2)(10x3-18x2-2x+8)* (x+3/2)(2)(5x3-9x2-x+4) -factor out GCF (2x+3)(5x3-9x2-x+4) -multiply 1st factor by 2

  12. Repeat finding zeros for:

  13. If the highest degree is more than 3 If the highest degree is more than 3 (like 4) you will need to do synthetic division again, this time on the “new” equation you just found. Your goal is to divide your equation down to a 2nd degree equation so you can factor or use the quadratic formula. Each time you do synthetic division, your equation goes down 1 degree.

  14. What is the rational zero theorem? If f(x)=anx + +a1x+a0 has integer coefficients, then every rational zero of f has the following form: pfactor of constant term a0 q factor of leading coefficient an What information does it give you? It gives you a pool of numbersto use to help you find a divisor. =

  15. Assignment 2.6 p. 132, 24-34 even, 41-43 all

More Related