a) 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 ) = 2 x 5 − 13 x 4 + 32 x 3 − 38 x 2 + 22 x − 5 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 5 and q is the factor of 2. The possibilities of p q are, Possibilities for p Possibilities for q : ± 1 , ± 5 ± 1 , ± 2 Thus, the possibilities for the rational zeros of the function f ( x ) = 2 x 5 − 13 x 4 + 32 x 3 − 38 x 2 + 22 x − 5 are ± 1 , ± 5 , ± 1 2 , ± 5 2 . Try for the above possibilities, if f ( x ) = 0 . Here, 1 satisfies the condition f ( 1 ) = 0 , hence 1 is a zero of f ( x ) . As 1 is one of the zero, it is possible to write the polynomial as, f ( x ) = ( x − 1 ) ⋅ Q ( x ) . Synthetic division is used to find the value of Q ( x ) . 2 2 − 13 32 − 38 22 − 5 2 − 11 21 − 17 5 _ 2 − 11 21 − 17 5 0 Thus, the value of Q ( x ) = ( 2 x 4 − 11 x 3 + 21 x 2 − 17 x + 5 ) . Here, 1 satisfies the condition Q ( 1 ) = 0 , hence 1 is a zero of Q ( x ) . Then, the polynomial Q ( x ) becomes, Q ( x ) = ( x − 1 ) ⋅ R ( x ) . Use the synthetic division to find the value of R ( x ) b) To determine The factor of the polynomial into linear factors.

6th Edition

ISBN: 9780135962312

Author: BITTINGER

Publisher: PEARSON

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 4, Problem 66RE

To determine

The possible rational zeros and the other zeros of the function f(x)=2x5−13x4+32x3−38x2+22x−5.

To determine

The factor of the polynomial into linear factors.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4, Problem 65RE

Chapter 4, Problem 67RE

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 possible rational zeros and the other zeros of the function f ( x ) = 2 x 5 − 13 x 4 + 32 x 3 − 38 x 2 + 22 x − 5 .

Solution Summary: The author explains the rational zeros of the polynomial function f(x): p and q are relatively prime.