7) Using the fact that x = 1 is a zero of the polynomial P(x) = x³ - 4x² - 7x + 10, reduce P(x) to a degree 2 polynomial, and then find the other two zeros.
7) Using the fact that x = 1 is a zero of the polynomial P(x) = x³ - 4x² - 7x + 10, reduce P(x) to a degree 2 polynomial, and then find the other two zeros.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Polynomial Reduction and Finding Zeros
**Problem Statement:**
Using the fact that \( x = 1 \) is a zero of the polynomial \( P(x) = x^3 - 4x^2 - 7x + 10 \), reduce \( P(x) \) to a degree 2 polynomial, and then find the other two zeros.
**Detailed Solution:**
1. **Identify the given zero:**
Since \( x = 1 \) is a zero of the polynomial \( P(x) \), we know that \( (x - 1) \) is a factor of \( P(x) \).
2. **Use synthetic division to reduce the polynomial:**
Perform synthetic division to divide \( P(x) \) by \( (x - 1) \):
- Start by writing the coefficients of \( P(x) \):
\[
1, -4, -7, 10
\]
- Use \( x = 1 \) as the zero:
\[
\begin{array}{r|cccc}
1 & 1 & -4 & -7 & 10 \\
\hline
& & 1 & -3 & -10 \\
\hline
& 1 & -3 & -10 & 0 \\
\end{array}
\]
After synthetic division, we get the quotient polynomial:
\[
x^2 - 3x - 10
\]
3. **Factor the resulting quadratic polynomial:**
Find the factors of \( x^2 - 3x - 10 \).
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
4. **Determine the zeros of the quadratic polynomial:**
Set each factor equal to zero and solve for \( x \):
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
Therefore, the zeros of the original polynomial \( P(x) \) are:
- \( x = 1 \) (given)
- \( x = 5 \)
- \( x = -2 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e61a896-eac2-4e92-aac1-4d7f245a1e73%2Fe1fca551-6f4f-4002-aab2-77487498d273%2F667k6g6.jpeg&w=3840&q=75)
Transcribed Image Text:### Polynomial Reduction and Finding Zeros
**Problem Statement:**
Using the fact that \( x = 1 \) is a zero of the polynomial \( P(x) = x^3 - 4x^2 - 7x + 10 \), reduce \( P(x) \) to a degree 2 polynomial, and then find the other two zeros.
**Detailed Solution:**
1. **Identify the given zero:**
Since \( x = 1 \) is a zero of the polynomial \( P(x) \), we know that \( (x - 1) \) is a factor of \( P(x) \).
2. **Use synthetic division to reduce the polynomial:**
Perform synthetic division to divide \( P(x) \) by \( (x - 1) \):
- Start by writing the coefficients of \( P(x) \):
\[
1, -4, -7, 10
\]
- Use \( x = 1 \) as the zero:
\[
\begin{array}{r|cccc}
1 & 1 & -4 & -7 & 10 \\
\hline
& & 1 & -3 & -10 \\
\hline
& 1 & -3 & -10 & 0 \\
\end{array}
\]
After synthetic division, we get the quotient polynomial:
\[
x^2 - 3x - 10
\]
3. **Factor the resulting quadratic polynomial:**
Find the factors of \( x^2 - 3x - 10 \).
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
4. **Determine the zeros of the quadratic polynomial:**
Set each factor equal to zero and solve for \( x \):
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
Therefore, the zeros of the original polynomial \( P(x) \) are:
- \( x = 1 \) (given)
- \( x = 5 \)
- \( x = -2 \)
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