Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 73

Chapter 4, Problem 73

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. 2-i and 6-3i

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Identify the given zeros: $2 - i$ and $6 - 3i$. Since the polynomial must have real coefficients, include their complex conjugates as zeros as well: $2 + i$ and $6 + 3i$.

Write the factors corresponding to each zero. For zero $a$, the factor is $(x - a)$. So the factors are: $(x - (2 - i))$, $(x - (2 + i))$, $(x - (6 - 3i))$, and $(x - (6 + 3i))$.

Group the conjugate pairs and multiply each pair to get quadratic factors with real coefficients: multiply $(x - (2 - i))$ and $(x - (2 + i))$ to get one quadratic factor, and multiply $(x - (6 - 3i))$ and $(x - (6 + 3i))$ to get the other quadratic factor.

Use the formula for multiplying conjugate binomials: $(x - (a - bi))(x - (a + bi)) = (x - a)^2 + b^2$. Apply this to each pair to simplify the quadratic factors.

Multiply the two quadratic factors obtained in the previous step to get the polynomial function $f(x)$ of least degree with real coefficients and the given zeros.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Conjugate Root Theorem

For polynomials with real coefficients, any non-real complex roots must occur in conjugate pairs. This means if 2 - i is a root, then its conjugate 2 + i must also be a root to ensure the polynomial has real coefficients.

Forming Polynomial from Roots

A polynomial can be constructed by creating factors from each root in the form (x - root). Multiplying these factors together yields a polynomial with the given roots. For example, roots r1 and r2 give factors (x - r1)(x - r2).

Simplifying Polynomial with Complex Roots

When multiplying factors involving complex conjugates, the product results in a quadratic with real coefficients. For instance, (x - (2 - i))(x - (2 + i)) simplifies to a quadratic with real coefficients, eliminating imaginary parts.

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Complex Conjugates

Related Practice

Textbook Question

Height of an Object If an object is projected upward from an initial height of 100 ft with an initial velocity of 64 ft per sec, then its height in feet after t seconds is given by $s (t) = - 16 t^{2} + 64 t + 100$ . Find the number of seconds it will take the object to reach its maximum height. What is this maximum height?

Textbook Question

Solve each problem. Work each of the following. Sketch the graph of a function that does not intersect its horizontal asymptote y=1, has the line x=3 as a vertical asymptote, and has x-intercepts (2, 0) and (4, 0).

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Textbook Question

Graph each rational function. ƒ(x)=(3x²+3x-6)/(x²-x-12)

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Textbook Question

Solve each problem. Work each of the following. Find an equation for a possible corresponding rational function.

Textbook Question

Graph each rational function. See Examples 5–9.

$ƒ (x) = [(x + 6) (x - 2)] / [(x + 3) (x - 4)]$

Textbook Question

Graph each rational function. ƒ(x)=[(x+3)(x-5)]/[(x+1)(x-4)]