Consider the following. P(x) = x³ − 9x² + 27x − 27 Use the remainder theorem to determine which is a zero of P, x = 3 or x = 4? x = 3 is a zero of P. x = 4 is a zero of P. Factor the polynomial as a product of linear factors with complex coefficients.
Consider the following. P(x) = x³ − 9x² + 27x − 27 Use the remainder theorem to determine which is a zero of P, x = 3 or x = 4? x = 3 is a zero of P. x = 4 is a zero of P. Factor the polynomial as a product of linear factors with complex coefficients.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Polynomial Analysis and Factorization
#### Consider the following polynomial:
\[ P(x) = x^3 - 9x^2 + 27x - 27 \]
#### Use the Remainder Theorem to determine which is a zero of \( P \):
- \( x = 3 \)
- \( x = 4 \)
**Explanation:**
- If \( x = 3 \) is substituted into the polynomial \( P(x) \), and it results in zero, then \( x = 3 \) is a zero of \( P \).
- If \( x = 4 \) is substituted into the polynomial \( P(x) \), and it doesn't result in zero, then \( x = 4 \) is not a zero of \( P \).
**Result:**
- \( x = 3 \) is a zero of \( P \). (This is indicated by the blue filled radio button and a green check mark.)
- \( x = 4 \) is not a zero of \( P \). (The radio button for \( x = 4 \) is unselected.)
#### Factor the polynomial as a product of linear factors with complex coefficients:
\[ P(x) \neq (x - 3)^3 \]
**Explanation:**
- The factorization \( P(x) = (x - 3)^3 \) is not correct for this polynomial. This is indicated by a red cross.
By accurately determining the zeros and correctly factoring the polynomial, one can better understand its properties and behavior. For a more detailed analysis, further algebraic steps or numerical methods may be necessary to find the correct factorization.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe45f8775-1248-4605-b15f-ac47ffaa832c%2F11d12fec-5878-4d19-b9bc-62b56f11ef83%2F8cnh33e_processed.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Analysis and Factorization
#### Consider the following polynomial:
\[ P(x) = x^3 - 9x^2 + 27x - 27 \]
#### Use the Remainder Theorem to determine which is a zero of \( P \):
- \( x = 3 \)
- \( x = 4 \)
**Explanation:**
- If \( x = 3 \) is substituted into the polynomial \( P(x) \), and it results in zero, then \( x = 3 \) is a zero of \( P \).
- If \( x = 4 \) is substituted into the polynomial \( P(x) \), and it doesn't result in zero, then \( x = 4 \) is not a zero of \( P \).
**Result:**
- \( x = 3 \) is a zero of \( P \). (This is indicated by the blue filled radio button and a green check mark.)
- \( x = 4 \) is not a zero of \( P \). (The radio button for \( x = 4 \) is unselected.)
#### Factor the polynomial as a product of linear factors with complex coefficients:
\[ P(x) \neq (x - 3)^3 \]
**Explanation:**
- The factorization \( P(x) = (x - 3)^3 \) is not correct for this polynomial. This is indicated by a red cross.
By accurately determining the zeros and correctly factoring the polynomial, one can better understand its properties and behavior. For a more detailed analysis, further algebraic steps or numerical methods may be necessary to find the correct factorization.
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