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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]

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)
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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**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: \[ x^3 - 5x^2 + 4x - 20 \]
Transcribed Image Text:**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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