Skip to main content
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 9
Chapter 4, Problem 9

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x3+x2−4x−4

Verified step by step guidance
1
Identify the polynomial function: \(f(x) = x^{3} + x^{2} - 4x - 4\).
List all possible rational zeros using the Rational Root Theorem. The possible rational zeros are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (-4) and \(q\) divides the leading coefficient (1). So, possible zeros are \(\pm1\), \(\pm2\), and \(\pm4\).
Use synthetic division to test each possible rational zero. Start by setting up synthetic division with one of the possible zeros (for example, \(x = 1\)) and divide the polynomial to see if the remainder is zero.
Once you find a zero that gives a remainder of zero, that value is an actual zero of the polynomial. The quotient from the synthetic division will be a quadratic polynomial.
Use the quadratic formula or factoring to find the remaining zeros from the quadratic quotient obtained in the previous step.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Rational Root Theorem

The Rational Root Theorem helps identify all possible rational zeros of a polynomial by considering factors of the constant term and the leading coefficient. These possible roots are expressed as ±(factors of constant term)/(factors of leading coefficient), providing a finite list to test.
Recommended video:
Guided course
04:06
Rational Exponents

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form (x - c). It simplifies the process of evaluating whether a candidate root is an actual zero by checking if the remainder is zero, and it produces a quotient polynomial for further factorization.
Recommended video:
05:10
Higher Powers of i

Factoring Polynomials and Finding Zeros

Once a zero is found, the polynomial can be factored using the quotient from synthetic division. This reduced polynomial is then solved to find remaining zeros, which may be real or complex, completing the factorization and root-finding process.
Recommended video:
Guided course
07:30
Introduction to Factoring Polynomials
Related Practice
Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.


As x→−3−x\(\to\)-3^{-}, f(x)→f\(\left\)(x\(\right\))\(\to\)____

643
views
Textbook Question

Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is the minimum product?

1379
views
1
comments
Textbook Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2(x−3)2+1

1115
views
Textbook Question

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as a and b and inversely as the square root of c. y = 12 when a = 3, b = 2, and c = 25. Find y when a = 5, b = 3 and c = 9.

644
views
Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x2−6x+9<0

499
views
Textbook Question

Divide using long division. State the quotient, and the remainder, r(x)r(x). 2x3+7x2+9x−20x+3\(\frac{2x^3 + 7x^2 + 9x - 20}{x + 3}\)

733
views