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 33
Chapter 4, Problem 33

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x3−11x2+7x−5;f(4)

Verified step by step guidance
1
Identify the polynomial function and the value at which you need to evaluate it: \(f(x) = 2x^{3} - 11x^{2} + 7x - 5\) and you want to find \(f(4)\).
Set up synthetic division by writing the coefficients of the polynomial in order: 2, -11, 7, -5. Use the value 4 as the divisor (since you are evaluating at \(x=4\)).
Begin synthetic division by bringing down the first coefficient (2) as is. Then multiply this number by 4 and write the result under the next coefficient.
Add the numbers in the second column, then repeat the multiply and add process for the remaining coefficients.
The final number you get after completing synthetic division is the remainder, which equals \(f(4)\) by the Remainder Theorem.

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 it faster and less error-prone, especially for evaluating polynomials at specific values.
Recommended video:
05:10
Higher Powers of i

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This means evaluating the polynomial at x = c gives the remainder directly, which is useful for quickly finding function values without full polynomial evaluation.
Recommended video:
05:10
Higher Powers of i

Polynomial Evaluation

Polynomial evaluation involves finding the value of a polynomial function at a specific input. Using synthetic division or direct substitution, one can efficiently compute f(c) for a given c, which is essential for understanding function behavior and solving related algebra problems.
Recommended video:
Guided course
05:13
Introduction to Polynomials
Related Practice
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. (2−x)2(x−7/2)<0

757
views
Textbook Question

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=(x+7)/(x2+4x−21)

1051
views
Textbook Question
Given f(x) = 2x^3 - 7x^2 + 9x - 3, use the Remainder Theorem to find f(- 13).
1134
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. x(4−x)(x−6)≤0

496
views
Textbook Question

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=x3+2x2+5x+4

493
views
Textbook Question

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=x2+6x+3

608
views