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

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on understanding marketing management Ppt on child labour download Human eye anatomy and physiology ppt on cells Download ppt on web designing Ppt on marie curie inventions Mba ppt on body language Ppt on power system stability using facts devices Ppt on refraction and reflection of light Ppt on water pollution in hindi language Ppt on personality development and grooming