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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 36
Chapter 4, Problem 36

For each polynomial function, one zero is given. Find all other zeros. ƒ(x)=4x3+6x2−2x−1; 12ƒ(x)=4x^3+6x^2-2x-1;\(\text{ }\]\frac\)12

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1
Identify the given polynomial function: \(f(x) = 4x^3 + 6x^2 - 2x - 1\), and the known zero: \(x = \frac{1}{2}\).
Use polynomial division or synthetic division to divide \(f(x)\) by the factor corresponding to the known zero, which is \(\left(x - \frac{1}{2}\right)\).
Perform the division to find the quotient polynomial, which will be a quadratic polynomial since the original is cubic.
Set the quotient polynomial equal to zero and solve for the remaining zeros using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are coefficients from the quadratic.
List all zeros: the known zero \(\frac{1}{2}\) and the two zeros found from solving the quadratic equation.

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Key Concepts

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

Polynomial Zeros and Roots

Zeros or roots of a polynomial are the values of x that make the polynomial equal to zero. Finding all zeros involves solving the polynomial equation, which can include real and complex solutions. Knowing one zero helps in factoring the polynomial to find the others.
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Imaginary Roots with the Square Root Property

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a binomial corresponding to a known zero, simplifying the polynomial to a lower degree. Synthetic division is a shortcut method when dividing by linear factors, making it easier to find remaining zeros.
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Introduction to Factoring Polynomials

Factoring and Solving Quadratic Equations

After dividing the polynomial, the quotient is often a quadratic that can be factored or solved using the quadratic formula. This step is essential to find the remaining zeros once the polynomial is reduced to degree two.
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Related Practice
Textbook Question

Solve each problem. Is x+1 a factor of ƒ(x)=x3+2x2+3x+2?

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Solve each problem. The speed of a pulley varies inversely as its diameter. One kind of pulley, with diameter 3 in., turns at 150 revolutions per minute. Find the speed of a similar pulley with diameter 5 in.

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Textbook Question

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = x3 - 4x2 + 2x+1; k = -1

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=(4x+3)(x+2)2

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=(3x-1)(x+2)2

Textbook Question

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = - x3 + 8x2 + 63; k=4