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}}

Ch. 3 - Polynomial and Rational Functions

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Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

문제 17

4장, 문제 17

Use synthetic division to perform each division. $\frac{\frac{1}{3} x^{3} - \frac{2}{9} x^{2} + \frac{2}{27} x - \frac{1}{81}}{x - \frac{1}{3}}$

검증된 단계별 안내

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 $.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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