### Complex Roots and Polynomial Equations **Problem 12: Polynomial Roots Verification** Consider the polynomial function defined as: \[ p(x) = x^3 - 5x^2 + 4x - 20 \] 1. **Verify that \( p(5) = 0 \)**: - This involves substituting \( x = 5 \) into the polynomial and confirming that the result is zero. 2. **Find the other roots of \( p(x) = 0 \) over the complex numbers**: - Once it is shown that \( x = 5 \) is a root, factor \( p(x) \) accordingly and solve the remaining polynomial equation to find all complex roots. #### Solution Given \( p(x) = x^3 - 5x^2 + 4x - 20 \): a. **Verify \( p(5) = 0 \)**: - Substituting \( x = 5 \) into \( p(x) \): \[ p(5) = (5)^3 - 5(5)^2 + 4(5) - 20 \] \[ = 125 - 125 + 20 - 20 \] \[ = 0 \] b. **Finding the other roots**: - With \( x = 5 \) being a root, factor \( p(x) \) as follows: \[ p(x) = (x - 5)(x^2 + bx + c) \] - By polynomial division or synthetic division, find: \[ b = 0 \] \[ c = -4 \] - Thus: \[ p(x) = (x - 5)(x^2 + 4) \] - Solve \( x^2 + 4 = 0 \): \[ x^2 = -4 \] \[ x = \pm 2i \] So, the other roots of \( p(x) \) are \( 2i \) and \( -2i \). For complete clarity in solving and verifying the roots, further break down each step on separate sections or pages, providing detailed explanations and visual aids if necessary (e.g., graphs showcasing the polynomial function and its roots). **Answer:** The other roots of \( p(x) \) are \( \boxed{
### Complex Roots and Polynomial Equations **Problem 12: Polynomial Roots Verification** Consider the polynomial function defined as: \[ p(x) = x^3 - 5x^2 + 4x - 20 \] 1. **Verify that \( p(5) = 0 \)**: - This involves substituting \( x = 5 \) into the polynomial and confirming that the result is zero. 2. **Find the other roots of \( p(x) = 0 \) over the complex numbers**: - Once it is shown that \( x = 5 \) is a root, factor \( p(x) \) accordingly and solve the remaining polynomial equation to find all complex roots. #### Solution Given \( p(x) = x^3 - 5x^2 + 4x - 20 \): a. **Verify \( p(5) = 0 \)**: - Substituting \( x = 5 \) into \( p(x) \): \[ p(5) = (5)^3 - 5(5)^2 + 4(5) - 20 \] \[ = 125 - 125 + 20 - 20 \] \[ = 0 \] b. **Finding the other roots**: - With \( x = 5 \) being a root, factor \( p(x) \) as follows: \[ p(x) = (x - 5)(x^2 + bx + c) \] - By polynomial division or synthetic division, find: \[ b = 0 \] \[ c = -4 \] - Thus: \[ p(x) = (x - 5)(x^2 + 4) \] - Solve \( x^2 + 4 = 0 \): \[ x^2 = -4 \] \[ x = \pm 2i \] So, the other roots of \( p(x) \) are \( 2i \) and \( -2i \). For complete clarity in solving and verifying the roots, further break down each step on separate sections or pages, providing detailed explanations and visual aids if necessary (e.g., graphs showcasing the polynomial function and its roots). **Answer:** The other roots of \( p(x) \) are \( \boxed{
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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![### Complex Roots and Polynomial Equations
**Problem 12: Polynomial Roots Verification**
Consider the polynomial function defined as:
\[ p(x) = x^3 - 5x^2 + 4x - 20 \]
1. **Verify that \( p(5) = 0 \)**:
- This involves substituting \( x = 5 \) into the polynomial and confirming that the result is zero.
2. **Find the other roots of \( p(x) = 0 \) over the complex numbers**:
- Once it is shown that \( x = 5 \) is a root, factor \( p(x) \) accordingly and solve the remaining polynomial equation to find all complex roots.
#### Solution
Given \( p(x) = x^3 - 5x^2 + 4x - 20 \):
a. **Verify \( p(5) = 0 \)**:
- Substituting \( x = 5 \) into \( p(x) \):
\[ p(5) = (5)^3 - 5(5)^2 + 4(5) - 20 \]
\[ = 125 - 125 + 20 - 20 \]
\[ = 0 \]
b. **Finding the other roots**:
- With \( x = 5 \) being a root, factor \( p(x) \) as follows:
\[ p(x) = (x - 5)(x^2 + bx + c) \]
- By polynomial division or synthetic division, find:
\[ b = 0 \]
\[ c = -4 \]
- Thus:
\[ p(x) = (x - 5)(x^2 + 4) \]
- Solve \( x^2 + 4 = 0 \):
\[ x^2 = -4 \]
\[ x = \pm 2i \]
So, the other roots of \( p(x) \) are \( 2i \) and \( -2i \).
For complete clarity in solving and verifying the roots, further break down each step on separate sections or pages, providing detailed explanations and visual aids if necessary (e.g., graphs showcasing the polynomial function and its roots).
**Answer:** The other roots of \( p(x) \) are \( \boxed{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8bd614aa-ccd1-4f8f-a7be-f9e61d8e2422%2Fb65ef7ba-f52f-4935-a181-d0847c4afead%2F847oio.jpeg&w=3840&q=75)
Transcribed Image Text:### Complex Roots and Polynomial Equations
**Problem 12: Polynomial Roots Verification**
Consider the polynomial function defined as:
\[ p(x) = x^3 - 5x^2 + 4x - 20 \]
1. **Verify that \( p(5) = 0 \)**:
- This involves substituting \( x = 5 \) into the polynomial and confirming that the result is zero.
2. **Find the other roots of \( p(x) = 0 \) over the complex numbers**:
- Once it is shown that \( x = 5 \) is a root, factor \( p(x) \) accordingly and solve the remaining polynomial equation to find all complex roots.
#### Solution
Given \( p(x) = x^3 - 5x^2 + 4x - 20 \):
a. **Verify \( p(5) = 0 \)**:
- Substituting \( x = 5 \) into \( p(x) \):
\[ p(5) = (5)^3 - 5(5)^2 + 4(5) - 20 \]
\[ = 125 - 125 + 20 - 20 \]
\[ = 0 \]
b. **Finding the other roots**:
- With \( x = 5 \) being a root, factor \( p(x) \) as follows:
\[ p(x) = (x - 5)(x^2 + bx + c) \]
- By polynomial division or synthetic division, find:
\[ b = 0 \]
\[ c = -4 \]
- Thus:
\[ p(x) = (x - 5)(x^2 + 4) \]
- Solve \( x^2 + 4 = 0 \):
\[ x^2 = -4 \]
\[ x = \pm 2i \]
So, the other roots of \( p(x) \) are \( 2i \) and \( -2i \).
For complete clarity in solving and verifying the roots, further break down each step on separate sections or pages, providing detailed explanations and visual aids if necessary (e.g., graphs showcasing the polynomial function and its roots).
**Answer:** The other roots of \( p(x) \) are \( \boxed{
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