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 18
Chapter 4, Problem 18

Use synthetic division to find ƒ(2). ƒ(x)=x5+4x2-2x-4

Verified step by step guidance
1
Identify the polynomial function: \(f(x) = x^5 + 4x^2 - 2x - 4\). Note that some terms are missing (like \(x^4\), \(x^3\), and the constant term), so include them with zero coefficients for synthetic division: \(x^5 + 0x^4 + 0x^3 + 4x^2 - 2x - 4\).
Set up synthetic division using the value \(x = 2\). Write down the coefficients in order: \([1, 0, 0, 4, -2, -4]\) and place the number 2 to the left.
Bring down the first coefficient (1) as is. Multiply it by 2 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 last number you obtain after completing the synthetic division process is the remainder, which equals \(f(2)\) according to the Remainder Theorem.
Interpret the remainder as the value of the function at \(x=2\), which is \(f(2)\). This completes the process of finding \(f(2)\) using synthetic division.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
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 shortcut method for dividing a polynomial by a linear binomial of the form (x - c). It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone.
Recommended video:
05:10
Higher Powers of i

Evaluating Polynomials Using Synthetic Division

Synthetic division can be used to evaluate a polynomial at a specific value by dividing the polynomial by (x - c) and examining the remainder. The remainder from this division equals the value of the polynomial at x = c.
Recommended video:
Guided course
04:15
Multiply Polynomials Using the Distributive Property

Polynomial Coefficients and Missing Terms

When performing synthetic division, all powers of x must be accounted for, including those with zero coefficients. For example, if a term like x^4 or x^3 is missing, its coefficient is zero and must be included in the synthetic division setup.
Recommended video:
Guided course
05:16
Standard Form of Polynomials
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

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 - \(\frac{1}{3}\)}

532
views
Textbook Question

Use synthetic division to perform each division. (x4 - 3x3 - 4x2 + 12x) / x-2

525
views
Textbook Question

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

646
views
Textbook Question

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

568
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