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 26

Chapter 4, Problem 26

Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. -2+√5, -2-√5, -2, 1

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Identify the given zeros of the polynomial: $-2 + \sqrt{5}$, $-2 - \sqrt{5}$, $-2$, and $1$.

Since the polynomial has real coefficients, the conjugate pair $-2 + \sqrt{5}$ and $-2 - \sqrt{5}$ will form a quadratic factor. Write this factor as $\left(x - (-2 + \sqrt{5})\right)\left(x - (-2 - \sqrt{5})\right)$.

Simplify the quadratic factor by multiplying the conjugate binomials: $\left(x + 2 - \sqrt{5}\right)\left(x + 2 + \sqrt{5}\right)$, which can be expressed using the difference of squares formula.

Write the factors corresponding to the other zeros as linear factors: $\left(x - (-2)\right) = (x + 2)$ and $\left(x - 1\right)$.

Form the polynomial function $f(x)$ by multiplying all factors together: the quadratic factor from step 3 times the linear factors from step 4, resulting in a polynomial of least degree with real coefficients.

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Key Concepts

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

Polynomial Zeros and Factors

Each zero of a polynomial corresponds to a factor of the form (x - zero). To construct a polynomial with given zeros, multiply the factors associated with each zero. For example, a zero at c gives a factor (x - c).

Complex Conjugate Root Theorem

For polynomials with real coefficients, non-real or irrational roots occur in conjugate pairs. Since -2 + √5 and -2 - √5 are conjugates, both must be included to ensure the polynomial has real coefficients.

Constructing the Polynomial of Least Degree

The polynomial of least degree with given zeros is formed by multiplying the linear factors corresponding to each zero exactly once. This ensures the polynomial has the smallest possible degree while including all specified roots.

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Related Practice

Textbook Question

Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = (x - 5)² - 4

Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation. (x - 4)(2x + 3)(3x - 1) ≥ 0

Textbook Question

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=10x⁶-x⁵+2x-2

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Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. (2x - 1)(5x - 9)(x - 4) < 0

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Textbook Question

Factor ƒ(x) into linear factors given that k is a zero. $ƒ (x) = - 6 x^{3} - 25 x^{2} - 3 x + 4; k = - 4$

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Textbook Question

Use synthetic division to divide ƒ(x) by x-k for the given value of k. Then express ƒ(x) in the form ƒ(x) = (x-k) q(x) + r. ƒ(x) = 2x³ + 3x² - 16x+10; k = -4