**Problem 5: Polynomial Construction***Objective:* Find a polynomial with integer coefficients that has degree 3 and zeros 5, and $2i$. Multiply out to write this polynomial in standard form.\[ \text{_______________________________} \]**Instructions:**1. Recall that complex zeros occur in conjugate pairs if the polynomial has real coefficients. Thus, if $2i$ is a zero, then $-2i$ must also be a zero.2. Identify the zeros of the polynomial: $5$, $2i$, and $-2i$.3. Construct factors from the zeros: - For the zero 5: $ (x - 5) $ - For the zero $2i$: $ (x - 2i) $ - For the zero $-2i$: $ (x + 2i) $4. Multiply the factors to construct the polynomial: \[ (x - 5)(x - 2i)(x + 2i) \]5. Simplify the expression: - First, multiply the complex conjugate pair: \[ (x - 2i)(x + 2i) = x^2 - (2i)^2 = x^2 + 4 \] - Next, multiply by the remaining factor: \[ (x - 5)(x^2 + 4) = x(x^2 + 4) - 5(x^2 + 4) \] - Distribute and combine like terms: \[ x^3 + 4x - 5x^2 - 20 = x^3 - 5x^2 + 4x - 20 \]The polynomial in standard form is:\[ x^3 - 5x^2 + 4x - 20 \]

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Answered: 5. Find a polynomial with integer…

5. Find a polynomial with integer coefficients that has degree 3 and zeros 5, and 2i. [Multiply out to write this polynomial in standard form]

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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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.

Question

**Problem 5: Polynomial Construction**

*Objective:* Find a polynomial with integer coefficients that has degree 3 and zeros 5, and $2i$. Multiply out to write this polynomial in standard form.

\[ \text{_______________________________} \]

**Instructions:**
1. Recall that complex zeros occur in conjugate pairs if the polynomial has real coefficients. Thus, if $2i$ is a zero, then $-2i$ must also be a zero.
2. Identify the zeros of the polynomial: $5$, $2i$, and $-2i$.
3. Construct factors from the zeros:
- For the zero 5: $ (x - 5) $
- For the zero $2i$: $ (x - 2i) $
- For the zero $-2i$: $ (x + 2i) $
4. Multiply the factors to construct the polynomial:
\[
(x - 5)(x - 2i)(x + 2i)
\]
5. Simplify the expression:
- First, multiply the complex conjugate pair:
\[
(x - 2i)(x + 2i) = x^2 - (2i)^2 = x^2 + 4
\]
- Next, multiply by the remaining factor:
\[
(x - 5)(x^2 + 4) = x(x^2 + 4) - 5(x^2 + 4)
\]
- Distribute and combine like terms:
\[
x^3 + 4x - 5x^2 - 20 = x^3 - 5x^2 + 4x - 20
\]

The polynomial in standard form is:
\[ x^3 - 5x^2 + 4x - 20 \]

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