Math Lesson2.notebook 1 October 31, 2013 5.6 Find Rational Zeros 10/31 Rational Zero Theorem: If f(x) = a n x n + ... + 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 q factor of leading coefficient a n Ex: List all possible zeros a) f(x) = x 3 + 2x 2 11x + 12 b) f(x) = 4x 4 x 3 3x 2 + 9x 10
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Math Lesson2.notebook
1
October 31, 2013
5.6 Find Rational Zeros 10/31
Rational Zero Theorem: If f(x) = anxn + ... + a1x + a0 has integer coefficients, then every rational zero of f has the following form:
p = factor of constant term a0 q factor of leading coefficient an
Ex: List all possible zerosa) f(x) = x3 + 2x2 11x + 12
b) f(x) = 4x4 x3 3x2 + 9x 10
Math Lesson2.notebook
2
October 31, 2013
Finding Zeros when the leading coefficient is 1: Step 1 List the possible rational zeros.Step 2 Test the zeros using synthetic
division.Step 3 Factor the trinomial in f(x) and use
the factor theorem.
Ex: f(x) = x3 8x2 + 11x + 20
Math Lesson2.notebook
3
October 31, 2013
Find zeros when the leading coefficient isn't 1:Step 1: List possible rational zeros.Step 2: Choose a reasonable value & check by
graphing.Step 3: Check values by synthetic division.Step 4: Factor out the binomial.Step 5: Repeat above steps from above.
(rational zeros will be the same)Step 6: Find remaining zeros.
Ex: f(x) = 10x4 11x3 42x2 + 7x + 12
Math Lesson2.notebook
4
October 31, 2013
NameOct. 31, 2013
HourP.374 #814e, 2022e
List the possible rational zeros using the rational zero theorem.8. f(x) = 3x4 + 5x3 3x + 4210. h(x) = 6x3 3x2 + 12Find all real zeros of the function.12. f(x) = x3 5x2 22x + 5614. h(x) = x3 + 8x2 9x 72Find all real zeros of the function. Use the graph to shorten your list. (graphs are in your book)20. f(x) = 4x3 12x2 x + 1522. f(x) = 3x3 + 20x2 36x + 16
Math Lesson2.notebook
5
October 31, 2013
Oct. 31, 2013P.374 #422e
List the possible rational zeros using the rational zero theorem.10. h(x) = 6x3 3x2 + 12
Factors of 12 are ±1, ±2,±3,±4, ±6, ±12Factors of 6 are ±1, ±2, ±3, ±6
Use Rational Zero Theorem:1/1, 2/1, 3/1, 4/1, 6/1, 12/1, 1/2, 2/2, 3/2, 4/2, 6/2, 12/2, 1/3, 2/3, 3/3, 4/3, 6/3, 12/3, 1/6, 2/6, 3/6, 4/6, 6/6, 12/6,
Eliminate repeat answers.±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2, ±1/3, ±2/3, ±4/3, ±1/6
Find all real zeros of the function.14. h(x) = x3 + 8x2 9x 72
Possible zeros are:±1, ±2, ±3, ±4, ±6, ±8, ±9, ±12, ±18, ±24, ±36, ±72
Test x = 1 Test x = 11 1 8 9 72 1 1 8 9 72
1 9 0 1 7 161 9 0 72 1 7 16 56
Test x = 2 Test x = 22 1 8 9 72 2 1 8 9 72
2 20 22 2 12 421 10 11 50 1 6 21 30
Test x = 33 1 8 9 72
3 33 72
1 11 24 0 ⇒ (x 3)(x2 + 11x + 24)Finish factoring.
(x 3)(x + 3)(x + 8)
Related Documents
