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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 25
Chapter 4, Problem 25

Factor ƒ(x) into linear factors given that k is a zero. ƒ(x)=−6x3−25x2−3x+4; k=−4ƒ(x)=-6x^3-25x^2-3x+4;\(\text{ }\)k=-4

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1
First, recognize that since k = -4 is a zero of the polynomial ƒ(x), it means that (x - (-4)) or (x + 4) is a factor of ƒ(x).
Use polynomial division or synthetic division to divide ƒ(x) = -6x^3 - 25x^2 - 3x + 4 by (x + 4). This will give you a quadratic quotient.
Set up synthetic division with -4 as the divisor and the coefficients of ƒ(x): -6, -25, -3, and 4.
Perform the synthetic division step-by-step to find the quotient polynomial, which will be a quadratic expression.
Once you have the quadratic quotient, factor it further into linear factors if possible, so that ƒ(x) is expressed as a product of linear factors including (x + 4).

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Key Concepts

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

Polynomial Zeros and the Factor Theorem

The Factor Theorem states that if k is a zero of a polynomial ƒ(x), then (x - k) is a factor of ƒ(x). This means substituting k into ƒ(x) yields zero, confirming (x - k) divides the polynomial exactly.
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Introduction to Factoring Polynomials

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide the original polynomial by the factor (x - k) to reduce its degree. Synthetic division is a shortcut method for dividing by linear factors, simplifying the process of factoring higher-degree polynomials.
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Introduction to Factoring Polynomials

Factoring Polynomials into Linear Factors

After dividing out the known factor, the resulting polynomial can be further factored into linear factors by finding its zeros. Fully factoring a cubic polynomial means expressing it as a product of three linear factors, if possible.
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Related Practice
Textbook Question

Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. √3, -√3, 2, 3

Textbook Question

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

Textbook Question

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=9x6-3x4+x2-2

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

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