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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 22
Chapter 4, Problem 22

Factor ƒ(x) into linear factors given that k is a zero. ƒ(x)=2x3−3x2−5x+6; k=1ƒ(x)=2x^3-3x^2-5x+6;^{\(\text{ }\)}k=1

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1
Since k = 1 is a zero of the polynomial ƒ(x) = 2x^3 - 3x^2 - 5x + 6, use synthetic division or polynomial division to divide ƒ(x) by (x - 1).
Set up synthetic division with the coefficients of ƒ(x): 2, -3, -5, and 6, and use 1 as the divisor.
Perform the synthetic division step-by-step to find the quotient polynomial, which will be a quadratic expression.
Write ƒ(x) as the product of (x - 1) and the quotient polynomial obtained from the division.
Factor the quadratic quotient further, if possible, into linear factors to express ƒ(x) completely as a product of linear factors.

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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

A zero of a polynomial is a value of x that makes the polynomial equal to zero. The Factor Theorem states that if k is a zero of ƒ(x), then (x - k) is a factor of ƒ(x). This allows us to factor the polynomial by dividing it by (x - k).
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Introduction to Factoring Polynomials

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a linear factor like (x - k). Synthetic division is a shortcut method for dividing by linear factors and helps find the quotient polynomial after factoring out (x - k). This simplifies the polynomial into lower-degree factors.
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Introduction to Factoring Polynomials

Factoring Polynomials into Linear Factors

After dividing out a known factor, the resulting polynomial can often be factored further into linear factors by finding its zeros. Factoring completely into linear factors means expressing the polynomial as a product of first-degree polynomials, which corresponds to all its roots.
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Related Practice
Textbook Question

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=-x3-4x2+2x-1

Textbook Question

Use one of the end behavior diagrams below, to describe the end behavior of the graph of each polynomial function.

ƒ(x)=−4x3+3x2−1 ƒ(x)=-4x^3+3x^2-1

Textbook Question

Write each formula as an English phrase using the word varies or proportional. r = d/t, where r is the speed when traveling d miles in t hours.

Textbook Question

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=5x5+2x3-3x+4

Textbook Question

Use synthetic division to perform each division. x4-1 / x-1

Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation.

(a) -x(x - 1)(x - 2) ≥ 0

(b) -x(x - 1)(x - 2) > 0

(c) -x(x - 1)(x - 2) ≤ 0

(d) -x(x - 1)(x - 2) < 0