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 as follows. Since the given polynomial is of degree 4, there are at most 4 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 8 and q is the factor of 1. The possibilities of p q are, Possibilities for p Possibilities for q : ± 1 , ± 2 , ± 4 , ± 8 ± 1 Thus, the possibilities for the rational zeros of the function f ( x ) = x 4 − 3 x 3 − 20 x 2 − 24 x − 8 are 1 , − 1 , 2 , − 2 , 4 , − 4 , 8 , − 8 . 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 − 20 − 24 − 8 − 2 10 20 8 _ 1 − 5 − 10 − 4 0 Thus, the value of Q ( x ) = ( x 4 − 5 x 2 − 10 x − 4 ) b) To determine The factor of the polynomial into linear factors.

6th Edition

ISBN: 9780135962312

Author: BITTINGER

Publisher: PEARSON

See similar textbooks

Concept explainers

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.

Videos

Question

Chapter 4.4, Problem 65E

To determine

The possible rational zeros and the other zeros of the function f(x)=x4−3x3−20x2−24x−8.

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 4.4, Problem 64E

Chapter 4.4, Problem 66E

Students have asked these similar questions

1) Express these large and small numbers from the Read and Study section in scientific notation: (a) 239,000 miles (b) 3,800,000,000,000 sheets of paper (c) 0.0000000000000000000000167 grams 2) Find all values for the variable x that make these equations true. (a) 5x = 1 (b) 3x = 1/1 9 (c) 4* = 11/ 4 (e) 4* = 64 (g) 10x = 1,000,000 (d) 3x=-3 (f) 2x = = 8 (h) 10x = 0.001

(b) 4) Find an equation to fit each of the following graphs: (a) 20 20 18 16 14 12 10 8 6 4 2 24 22 20 18 16 14 12 10 8 16 A 2 -3 -2 -1-0 2 3 4. -1 0 1 2 3. -2 -2

3) Which of the following are equivalent to 3? (There may be more than one that is equivalent!) -1 (a) (9)¯¹ 3. (b) (-3)-1 (c) (-3) -1 (d) -(¯3) (e) 11 3-1 (f) 3-4

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 ) = x 4 − 3 x 3 − 20 x 2 − 24 x − 8 .

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