Question

A polynomial $f(x)$ with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express $f(x)$ as a product of linear and quadratic polynomials with real coefficients that are irreducible over $\mathbb{R}$. $$3+2 i; \quad \text { degree } 2$$

Name: Problem 1, Chapter 3: Precalculus: Functions and Graphs
Uploaded: 2020-11-02T15:22:22-08:00
Duration: 58 s
Channel: Sriram Soundarrajan
Description: A polynomial f(x) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express f(x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over ℝ. 3+2 i; degree 2

   A polynomial $f(x)$ with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express $f(x)$ as a product of linear and quadratic polynomials with real coefficients that are irreducible over $\mathbb{R}$.
$$3+2 i; \quad \text { degree } 2$$

Precalculus: Functions and Graphs

Earl W. Swokowski,… 11th Edition

Chapter 3, Problem 1

Instant Answer

Solved by Expert Sriram Soundarrajan

11/02/2020

Step 1

We know that complex roots always come in conjugate pairs if the coefficients of the polynomial are real. Therefore, the other zero of the polynomial is the conjugate of $3+2i$, which is $3-2i$. Show more…

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A polynomial $f(x)$ with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express $f(x)$ as a product of linear and quadratic polynomials with real coefficients that are irreducible over $\mathbb{R}$. $$3+2 i; \quad \text { degree } 2$$

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Key Concepts

Complex Conjugate Root Theorem

When a polynomial has real coefficients, any non-real complex zeros must occur in conjugate pairs. This means that if a complex number a + bi is a zero, then its conjugate a - bi is also a zero. This property is essential when expressing polynomials with real coefficients into factors with real numbers only.

Factoring into Irreducible Real Polynomials

Using the complex conjugate pairs, we can construct irreducible quadratic factors with real coefficients by multiplying the linear factors for each conjugate pair. The product of (x - (a + bi)) and (x - (a - bi)) yields a quadratic polynomial of the form x² - 2ax + (a² + b²), which cannot be factored further over the real numbers.

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