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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 37
Chapter 4, Problem 37

For each polynomial function, one zero is given. Find all other zeros. ƒ(x)=−x4−5x2−4;−iƒ(x)=-x^4-5x^2-4; -i

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Identify the given polynomial function: \(f(x) = -x^{4} - 5x^{2} - 4\) and the given zero: \(-i\).
Recall that for polynomials with real coefficients, complex zeros come in conjugate pairs. Since \(-i\) is a zero, its conjugate \(i\) is also a zero.
Use the fact that \(x = i\) and \(x = -i\) are zeros to form a quadratic factor: \((x - i)(x + i) = x^{2} + 1\).
Divide the original polynomial \(f(x)\) by the quadratic factor \(x^{2} + 1\) using polynomial division or synthetic division to find the other quadratic factor.
Set the resulting quadratic factor equal to zero and solve for \(x\) to find the remaining zeros of the polynomial.

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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

This theorem states that if a polynomial has real coefficients and a complex number a + bi is a root, then its conjugate a - bi is also a root. Since -i is given as a zero, its conjugate i must also be a zero of the polynomial.
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Factoring Polynomials Using Known Roots

Once a root is known, the polynomial can be divided by the corresponding factor (x - root) to reduce its degree. Repeated use of this process helps find all zeros by factoring the polynomial completely.
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Polynomial Division and Synthetic Division

Polynomial division, including synthetic division, is a method to divide polynomials by linear factors. It simplifies the polynomial after factoring out known roots, making it easier to find remaining zeros.
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Related Practice
Textbook Question

Solve each problem. Find a polynomial function ƒ of degree 3 with -2, 1, and 4 as zeros, and ƒ(2)=16.

Textbook Question

Current Flow In electric current flow, it is found that the resistance offered by a fixed length of wire of a given material varies inversely as the square of the diameter of the wire. If a wire 0.01 in. in diameter has a resistance of 0.4 ohm, what is the resistance of a wire of the same length and material with diameter 0.03 in., to the nearest ten-thousandth of an ohm?

Textbook Question

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = x2 - 5x+1; k = 2+i

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Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = (x + 3)2

Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4.

ƒ(x)=x3+x2−36x−36ƒ(x)=x^3+x^2-36x-36

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=3/(x-5)

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