Download presentation

Presentation is loading. Please wait.

Published byKameron Drye Modified over 3 years ago

1
4.4 Finding Rational Zeros

2
The rational root 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 factors of constant term a 0 q factors of leading coefficient a n n … =

3
Example 1: Find rational roots of f(x)=x 3 +2x 2 -11x-12 1.List possible q=1(1) p=-12 (1,2,3,4,6,12) X= ±1/1,± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/1 2.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 3.Since -1 is a zero: (x+1)(x 2 +x-12)=f(x) Factor: (x+1)(x-3)(x+4)=0 x=-1 x=3 x=-4 You can use a graphing calculator to narrow your choices before you test!

4
Extra Example 1: Find rational zeros of: f(x)=x 3 -4x 2 -11x+30 1. q=1 (1) p=30 (1,2,3,5,6,10,30) x= ±1/1, ± 2/1, ±3/1, ±5/1, ±6/1, ±10/1, ±15/1, ±30/1 2.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)(x 2 -2x-15)=0 2 -4 -30 (x-2)(x+3)(x-5)=0 1 -2 -15 0 x=2 x=-3 x=5

5
Example 2: f(x)=10x 4 -3x 3 -29x 2 +5x+12 1.List: q=10 (1,2,5,10) p=12 (1,2,3,4,6,12) x= ± 1/1, ± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/1, ± ½, ± 3/2, ± 1/5, ± 2/5, ± 3/5, ± 4/5, ± 6/5, ± 12/5, ± 1/10, ± 3/10, ± 12/10 2.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)(10x 3 -18x 2 -2x+8)* (x+3/2)(2)(5x 3 -9x 2 -x+4) -factor out GCF (2x+3)(5x 3 -9x 2 -x+4) -multiply 1 st factor by 2 __ ____

6
Descartes’ Rule of Signs The number of positive real zeros = the number of sign changes in f(x) or less than by an even #. NO YES There are four changes. So there may be 4, 2 or 0 positive real zeros.

7
Descartes’ Rule of Signs cont. The number of negative real zeros = the number of sign changes in f(-x) or less than by an even #. YES NO There is one change. So there is 1 negative real zero.

8
Find the number of positive and negative real zeros. Then determine the rational zeros for g(x)= 5x 3 - 9x 2 – x + 4 There are 2 positive and 1 negative real zero 1. LC=5 CT=4 x:±1, ±2, ±4, ±1/5, ±2/5, ±4/5 *The graph of original shows 4/5 may be: 5 -9 -1 4 x=4/5 4 -4 -4 5 -5 -5 0 (x-4/5)(5x 2 -5x-5)= (x-4/5)(5)(x 2 -x-1)= mult.2 nd factor by 5 (5x-4)(x 2 -x-1)= -now use quad for last- * 4/5, 1+, 1-. 2 2 __ ____

9
AAAA ssss ssss iiii gggg nnnn mmmm eeee nnnn tttt: page 234 10,12,20,22

Similar presentations

OK

EXAMPLE 1 List possible rational zeros List the possible rational zeros of f using the rational zero theorem. a. f (x) = x 3 + 2x 2 – 11x + 12 Factors.

EXAMPLE 1 List possible rational zeros List the possible rational zeros of f using the rational zero theorem. a. f (x) = x 3 + 2x 2 – 11x + 12 Factors.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google