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
![**Problem 13:**
Find a fourth degree polynomial that has zeros of 1, -3, 2i, and -2i.
**Solution:**
To find a polynomial with the given zeros, we'll start with:
\[
(x - 1)(x + 3)(x - 2i)(x + 2i)
\]
- The expression \((x - 2i)(x + 2i)\) simplifies using the difference of squares:
\[
(x^2 - (2i)^2) = x^2 - (-4) = x^2 + 4
\]
- Then, multiply the first two factors:
\[
(x - 1)(x + 3) = x^2 + 3x - x - 3 = x^2 + 2x - 3
\]
- Therefore, the polynomial can be expanded as:
\[
(x^2 + 2x - 3)(x^2 + 4)
\]
- Multiply these two resulting polynomials:
\[
= x^4 + 4x^2 + 2x^3 + 8x - 3x^2 - 12
\]
- Combine like terms to get:
\[
= x^4 + 2x^3 + x^2 + 8x - 12
\]
Thus, the fourth degree polynomial is:
**Answer:**
\[ x^4 + 2x^3 + x^2 + 8x - 12 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa610010-7ad2-4d49-b949-5d449ad446df%2Fa7a57d65-a5d2-4a9a-abc5-ff33865048ba%2Fgp3vd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 13:**
Find a fourth degree polynomial that has zeros of 1, -3, 2i, and -2i.
**Solution:**
To find a polynomial with the given zeros, we'll start with:
\[
(x - 1)(x + 3)(x - 2i)(x + 2i)
\]
- The expression \((x - 2i)(x + 2i)\) simplifies using the difference of squares:
\[
(x^2 - (2i)^2) = x^2 - (-4) = x^2 + 4
\]
- Then, multiply the first two factors:
\[
(x - 1)(x + 3) = x^2 + 3x - x - 3 = x^2 + 2x - 3
\]
- Therefore, the polynomial can be expanded as:
\[
(x^2 + 2x - 3)(x^2 + 4)
\]
- Multiply these two resulting polynomials:
\[
= x^4 + 4x^2 + 2x^3 + 8x - 3x^2 - 12
\]
- Combine like terms to get:
\[
= x^4 + 2x^3 + x^2 + 8x - 12
\]
Thus, the fourth degree polynomial is:
**Answer:**
\[ x^4 + 2x^3 + x^2 + 8x - 12 \]
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