Skip to main content
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 17
Chapter 4, Problem 17

Use synthetic division to perform each division. 13x3−29x2+227x−181x−13\(\frac{\frac{1}{3}\)x^3 - \(\frac{2}{9}\)x^2 + \(\frac{2}{27}\)x - \(\frac{1}{81}\)}{x - \(\frac{1}{3}\)}

Verified step by step guidance
1
Identify the divisor and rewrite it in the form \( x - c \). Here, the divisor is \( x - \frac{1}{3} \), so \( c = \frac{1}{3} \).
List the coefficients of the dividend polynomial \( \frac{1}{3}x^3 - \frac{2}{9}x^2 + \frac{2}{27}x - \frac{1}{81} \). These are \( \frac{1}{3}, -\frac{2}{9}, \frac{2}{27}, -\frac{1}{81} \).
Set up the synthetic division by writing \( c = \frac{1}{3} \) on the left and the coefficients in a row to the right.
Bring down the first coefficient \( \frac{1}{3} \) as is. Then multiply it by \( c = \frac{1}{3} \) and write the result under the next coefficient. Add the column and write the sum below. Repeat this multiply-and-add process for all coefficients.
The numbers obtained after the last addition represent the coefficients of the quotient polynomial, and the final number is the remainder. Write the quotient polynomial using these coefficients with decreasing powers of \( x \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

Key Concepts

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

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form x - c. It uses only the coefficients of the polynomial and performs arithmetic operations in a tabular form, making the division process faster and less error-prone compared to long division.
Recommended video:
05:10
Higher Powers of i

Polynomial Coefficients and Terms

Understanding polynomial coefficients and terms is essential for synthetic division. Each term's coefficient must be correctly identified and arranged in descending order of degree, including zero coefficients for any missing powers, to ensure accurate computation during the division process.
Recommended video:
Guided course
05:16
Standard Form of Polynomials

Division by a Linear Binomial of the Form x - c

Synthetic division applies specifically when dividing by a linear binomial x - c. Recognizing the value of c (in this case, 1/3) is crucial because it is used in the synthetic division process to multiply and combine coefficients systematically to find the quotient and remainder.
Recommended video:
5:30
Foci and Vertices of an Ellipse
Related Practice
Textbook Question

Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. ƒ(x)=(1/3)(x+3)4-3

816
views
Textbook Question

Solve each quadratic inequality. Give the solution set in interval notation. x2 + x - 30 ≤ 0

568
views
Textbook Question

Match each statement with its corresponding graph in choices A–D. In each case, k > 0. y varies directly as x. (y=kx)

695
views
Textbook Question

Use synthetic division to find ƒ(2). ƒ(x)=5x4-12x2+2x-8

474
views
Textbook Question

Match each function with its graph without actually entering it into a calculator. Then, after completing the exercises, check the answers with a calculator. Use the standard viewing window. ƒ(x) = (x + 4)2 - 3

1000
views
Textbook Question

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. x3+2x2+3; x-1

828
views