Show that the binomial is a factor of the polynomial. Then factor the polynomial completely. s(x) = x + 4x – 64x – 256 ; x + 4 s(x) = D
Show that the binomial is a factor of the polynomial. Then factor the polynomial completely. s(x) = x + 4x – 64x – 256 ; x + 4 s(x) = D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 \( s(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe67f5cb0-daee-4ea8-9bfc-4a00f267029a%2F89db075f-41e8-4e6d-80ba-68143437c8db%2Fnhxm5cc_processed.jpeg&w=3840&q=75)
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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