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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 44
Chapter 4, Problem 44

Solve the equation 2x3−3x2−11x+6=0 given that -2 is a zero of f(x)=2x3−3x2−11x+6.

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Since -2 is a zero of the polynomial function \(f(x) = 2x^{3} - 3x^{2} - 11x + 6\), use synthetic division or polynomial long division to divide \(f(x)\) by the factor corresponding to this zero, which is \((x + 2)\).
Set up the synthetic division by writing the coefficients of \(f(x)\): 2, -3, -11, and 6. Then perform the division using -2 as the divisor.
After completing the synthetic division, write down the quotient polynomial, which will be a quadratic expression of the form \(ax^{2} + bx + c\).
Solve the quadratic equation obtained from the quotient by using factoring, completing the square, or the quadratic formula to find the remaining zeros of \(f(x)\).
Combine the zero \(x = -2\) with the solutions from the quadratic to write the complete solution set for the equation \(2x^{3} - 3x^{2} - 11x + 6 = 0\).

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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

A zero or root of a polynomial is a value of x that makes the polynomial equal to zero. Knowing a zero helps factor the polynomial and find other roots. For example, if -2 is a zero of f(x), then (x + 2) is a factor of f(x).
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Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a binomial factor, simplifying the polynomial. Synthetic division is a shortcut method when dividing by linear factors like (x - c). Dividing by (x + 2) will reduce the cubic to a quadratic.
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Factoring Quadratic Polynomials

After division, the resulting quadratic can be factored to find the remaining zeros. Factoring involves expressing the quadratic as a product of two binomials. If factoring is difficult, the quadratic formula can be used to find the roots.
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Related Practice
Textbook Question

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=x4−2x3+x2+12x+8

Textbook Question

Solve the equation 12x3+16x2−5x−3=0 given that -3/2 is a root.

Textbook Question

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. x4−3x3−20x2−24x−8=0

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Textbook Question

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=5x2−5x

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Textbook Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=(−2x+1)/(3x+5)

Textbook Question

Describe in words the variation shown by the given equation. z=kxy2z = \(\frac{k\sqrt{x}\)}{y^2}

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