Find the zeros and fully factor f(x) = x³ - 9x² + 22x - 10, including factors for irrational zeros. Use radicals, not decimal approximations. The zeros are The fully factor form is f(x) =
Find the zeros and fully factor f(x) = x³ - 9x² + 22x - 10, including factors for irrational zeros. Use radicals, not decimal approximations. The zeros are The fully factor form is f(x) =
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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Question
![**Title: Solving and Factoring Polynomials**
**Objective:**
In this lesson, we will learn how to find the zeros and fully factor the polynomial function \( f(x) = x^3 - 9x^2 + 22x - 10 \), including factors for irrational zeros. We will use radicals and not decimal approximations.
**Problem Statement:**
Find the zeros and fully factor \( f(x) = x^3 - 9x^2 + 22x - 10 \), including factors for irrational zeros. Use radicals, not decimal approximations.
**Answer Fields:**
- The zeros are: [___________]
- The fully factored form is \( f(x) = \) [___________]
**Explanation and Solution:**
To find the zeros of the polynomial, we look for the values of \( x \) for which \( f(x) = 0 \). Once we determine the zeros, which can be real or complex numbers, we can factor the polynomial accordingly.
**Graph or Diagram Description:**
There are no accompanying graphs or diagrams in the provided image. However, if creating a visual aid, it may be helpful to graph the polynomial function \( f(x) \) and observe where it intersects the x-axis to identify the real zeros.
**Steps for Solving:**
1. **Identify Possible Rational Zeros:** Use the Rational Root Theorem to list possible rational zeros.
2. **Synthetic Division:** Perform synthetic division to test potential zeros.
3. **Solve for Remaining Zeros:** Use the quadratic formula or other algebraic methods to find any remaining zeros if the polynomial is factored down to a quadratic.
4. **Fully Factor the Polynomial:** Express the polynomial in its factored form using the found zeros.
By following these steps, we can find the fully factored form of the polynomial \( f(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcae97c47-da87-4dbe-a1b5-55efe44ed72d%2F103e35ff-e1d1-45e6-b71f-3121c14d3b88%2Fcp344tt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving and Factoring Polynomials**
**Objective:**
In this lesson, we will learn how to find the zeros and fully factor the polynomial function \( f(x) = x^3 - 9x^2 + 22x - 10 \), including factors for irrational zeros. We will use radicals and not decimal approximations.
**Problem Statement:**
Find the zeros and fully factor \( f(x) = x^3 - 9x^2 + 22x - 10 \), including factors for irrational zeros. Use radicals, not decimal approximations.
**Answer Fields:**
- The zeros are: [___________]
- The fully factored form is \( f(x) = \) [___________]
**Explanation and Solution:**
To find the zeros of the polynomial, we look for the values of \( x \) for which \( f(x) = 0 \). Once we determine the zeros, which can be real or complex numbers, we can factor the polynomial accordingly.
**Graph or Diagram Description:**
There are no accompanying graphs or diagrams in the provided image. However, if creating a visual aid, it may be helpful to graph the polynomial function \( f(x) \) and observe where it intersects the x-axis to identify the real zeros.
**Steps for Solving:**
1. **Identify Possible Rational Zeros:** Use the Rational Root Theorem to list possible rational zeros.
2. **Synthetic Division:** Perform synthetic division to test potential zeros.
3. **Solve for Remaining Zeros:** Use the quadratic formula or other algebraic methods to find any remaining zeros if the polynomial is factored down to a quadratic.
4. **Fully Factor the Polynomial:** Express the polynomial in its factored form using the found zeros.
By following these steps, we can find the fully factored form of the polynomial \( f(x) \).
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