Use synthetic division to find ƒ(2). ƒ(x)=2x3-3x2+7x-12

Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 16

Chapter 4, Problem 16

Use synthetic division to find ƒ(2). ƒ(x)=2x³-3x²+7x-12

Verified step by step guidance

Identify the divisor for synthetic division. Since we want to find ƒ(2), the divisor is \( x - 2 \).

Write down the coefficients of the polynomial \( ƒ(x) = 2x^3 - 3x^2 + 7x - 12 \). These are \( 2, -3, 7, -12 \).

Set up synthetic division by placing the number 2 (from \( x - 2 \)) to the left and the coefficients in a row to the right.

Perform synthetic division by bringing down the first coefficient, multiplying it by 2, adding the result to the next coefficient, and repeating this process across all coefficients.

The final number obtained after completing synthetic division is the value of \( ƒ(2) \).

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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.

Evaluating Polynomials Using Synthetic Division

When dividing a polynomial by (x - c), the remainder of the synthetic division equals the value of the polynomial evaluated at x = c. This allows synthetic division to be used as a tool for quickly finding ƒ(c) without direct substitution.

Polynomial Coefficients and Setup

To perform synthetic division, list the coefficients of the polynomial in descending order of degree. If any terms are missing, include a zero coefficient. Proper setup ensures accurate computation of the division and evaluation.

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