**Degree 3 Polynomial Problem Set****Objective:**Find a degree 3 polynomial with real coefficients having zeros 1 and \( 4i \) and a lead coefficient of 1. Write \( P \) in expanded form. Be sure to write the full equation, including \( P(x) = \).**Given:**- Zeros: \( 1 \), \( 4i \)- Lead coefficient: 1**Instructions:**Write the polynomial \( P(x) \) in expanded form.**Polynomial Equation Format:**\[ P(x) = \]**Analysis of the Problem:**To construct this polynomial:1. Note that for a polynomial with real coefficients, any complex zeros must occur in conjugate pairs. Hence, if \( 4i \) is a zero, \( -4i \) must also be a zero.2. The polynomial having these three zeros \( 1 \), \( 4i \), and \( -4i \) can be written in factored form initially.3. Expand the factored form to express it in the standard polynomial form.**Steps:**1. The zeros \( 1 \), \( 4i \), and \( -4i \) correspond to factors \( (x-1) \), \( (x - 4i) \), and \( (x + 4i) \) respectively.2. Combine the complex conjugate pair factors first: \[ (x - 4i)(x + 4i) = x^2 - (4i)^2 = x^2 - (-16) = x^2 + 16 \]3. Multiply this result by the real factor \( (x-1) \): \[ P(x) = (x - 1)(x^2 + 16) \]4. Expand the product: \[ P(x) = x(x^2 + 16) - 1(x^2 + 16) = x^3 + 16x - x^2 - 16 \]5. Organize the polynomial in standard form: \[ P(x) = x^3 - x^2 + 16x - 16 \]**Final Polynomial:**\[ P(x) = x^3 - x^2 + 16x - 16 \]By following these steps, you have derived the polynomial

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Answered: Find a degree 3 polynomial with real…

Find a degree 3 polynomial with real coefficients having zeros 1 and 4i and a lead coefficient of 1. Write Pin expanded form. Be sure to write the full equation, including P(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

**Degree 3 Polynomial Problem Set**

**Objective:**
Find a degree 3 polynomial with real coefficients having zeros 1 and \( 4i \) and a lead coefficient of 1. Write \( P \) in expanded form. Be sure to write the full equation, including \( P(x) = \).

**Given:**
- Zeros: \( 1 \), \( 4i \)
- Lead coefficient: 1

**Instructions:**
Write the polynomial \( P(x) \) in expanded form.

**Polynomial Equation Format:**
\[ P(x) = \]

**Analysis of the Problem:**
To construct this polynomial:
1. Note that for a polynomial with real coefficients, any complex zeros must occur in conjugate pairs. Hence, if \( 4i \) is a zero, \( -4i \) must also be a zero.
2. The polynomial having these three zeros \( 1 \), \( 4i \), and \( -4i \) can be written in factored form initially.
3. Expand the factored form to express it in the standard polynomial form.

**Steps:**
1. The zeros \( 1 \), \( 4i \), and \( -4i \) correspond to factors \( (x-1) \), \( (x - 4i) \), and \( (x + 4i) \) respectively.
2. Combine the complex conjugate pair factors first:
\[ (x - 4i)(x + 4i) = x^2 - (4i)^2 = x^2 - (-16) = x^2 + 16 \]
3. Multiply this result by the real factor \( (x-1) \):
\[ P(x) = (x - 1)(x^2 + 16) \]

4. Expand the product:
\[ P(x) = x(x^2 + 16) - 1(x^2 + 16) = x^3 + 16x - x^2 - 16 \]

5. Organize the polynomial in standard form:
\[ P(x) = x^3 - x^2 + 16x - 16 \]

**Final Polynomial:**
\[ P(x) = x^3 - x^2 + 16x - 16 \]

By following these steps, you have derived the polynomial

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