Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Educational Exercise: Polynomial Factorization**
**Objective:** Show that the given binomial is a factor of the polynomial, then factor the polynomial completely.
**Problem Statement:**
Given the polynomial:
\[ s(x) = x^4 + 4x^3 - 64x - 256 \]
and the binomial:
\[ x + 4 \]
**Tasks:**
1. Demonstrate that \( x + 4 \) is a factor of \( s(x) \).
2. Factor the polynomial \( s(x) \) completely.
**Solution Approach:**
Start by using polynomial division or the Remainder Theorem to verify if \( x + 4 \) is a factor. If the remainder is zero, then \( x + 4 \) is a factor.
Proceed to factorize the polynomial completely by further breaking down the quotient obtained from the division process.
**Further Explanation:**
Explain each step of the division and factorization in detail, providing insights into methods used, such as synthetic division or long division for polynomials.
Conclude with a fully factored expression of \( s(x) \).](https://content.bartleby.com/qna-images/question/e67f5cb0-daee-4ea8-9bfc-4a00f267029a/89db075f-41e8-4e6d-80ba-68143437c8db/nhxm5cc_processed.jpeg)
Transcribed Image Text:**Educational Exercise: Polynomial Factorization**
**Objective:** Show that the given binomial is a factor of the polynomial, then factor the polynomial completely.
**Problem Statement:**
Given the polynomial:
\[ s(x) = x^4 + 4x^3 - 64x - 256 \]
and the binomial:
\[ x + 4 \]
**Tasks:**
1. Demonstrate that \( x + 4 \) is a factor of \( s(x) \).
2. Factor the polynomial \( s(x) \) completely.
**Solution Approach:**
Start by using polynomial division or the Remainder Theorem to verify if \( x + 4 \) is a factor. If the remainder is zero, then \( x + 4 \) is a factor.
Proceed to factorize the polynomial completely by further breaking down the quotient obtained from the division process.
**Further Explanation:**
Explain each step of the division and factorization in detail, providing insights into methods used, such as synthetic division or long division for polynomials.
Conclude with a fully factored expression of \( s(x) \).
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