Explanation Theorem used: “ The rational zeros theorem: Let P ( x ) = a n x n + a n − 1 x n − 1 + … + a 1 x + a 0 , where all the coefficients are integers. Consider a rational number denoted by p q , where p and q are relatively prime. If p q is a zero of P ( x ) , then p is a factor of a 0 and q is the factor of a n . Calculation: Obtain the possible rational zeros of the given function f ( x ) = x 5 − 3 x 4 − 3 x 3 + 9 x 2 − 4 x + 12 as follows. Since the given polynomial is of degree 5, there are at most 5 distinct zeros. From the rational zeros theorem, p q to be a zero of P ( x ) , p should be a factor of a 0 and q is the factor of a n . That is, p should be a factor of 12 and q is the factor of 1. The possibilities of p q are, Possibilities for p Possibilities for q : ± 1 , ± 2 , ± 3 , ± 4 , ± 6 , ± 12 ± 1 Thus, the possibilities for the rational zeros of the function f ( x ) = x 5 − 3 x 4 − 3 x 3 + 9 x 2 − 4 x + 12 are ± 1 , ± 2 , ± 3 , ± 4 , ± 6 and ± 12 . Try for the above possibilities, if f ( x ) = 0 . Here, –2 satisfies the condition f ( – 2 ) = 0 , hence –2 is a zero of f ( x ) . As –2 is one of the zero, it is possible to write the polynomial as, f ( x ) = ( x + 2 ) ⋅ Q ( x ) . Synthetic division is used to find the value of Q ( x ) . − 2 1 − 3 − 3 9 − 4 12 − 2 10 − 14 10 − 12 _ 1 − 5 7 − 5 6 0 Thus, the value of Q ( x ) = ( x 4 − 5 x 3 + 7 x 2 − 5 x + 6 ) . Here, 2 satisfies the condition Q ( 2 ) = 0 , hence 2 is a zero of Q ( x )

6th Edition

ISBN: 9780135962312

Author: BITTINGER

Publisher: PEARSON

See similar textbooks

Concept explainers

Roots

Videos

Question

Chapter 4.4, Problem 80E

To determine

The rational zeros of the function f(x)=x5−3x4−3x3+9x2−4x+12.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.4, Problem 79E

Chapter 4.4, Problem 81E

Students have asked these similar questions

I want a mathematical relationship with all the details, not explanations and definitions

4 sinx cos2x+4 cos x sin2x-1=0

For the matrix A, find (if possible) a nonsingular matrix P such that P-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) 6 -2 -[47] A = -3 1 P = Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. P-1AP =

Chapter 4 Solutions

COLL.ALG.:GRAPHS+MODELS-MYLABMATH ACC

Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 2E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 10E

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

The rational zeros of the function f ( x ) = x 5 − 3 x 4 − 3 x 3 + 9 x 2 − 4 x + 12 .

Solution Summary: The author explains the rational zeros of the polynomial function f(x).

Concept explainers

Videos

The rational zeros of the function f(x)=x5−3x4−3x3+9x2−4x+12.

Want to see the full answer?

Chapter 4 Solutions