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

Ch. 3 - Polynomial and Rational Functions

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

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Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 22

Chapter 4, Problem 22

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

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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).

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.

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

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Textbook Question

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

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