Divide using synthetic division. (x4−256)/(x−4)

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 29

4장, 문제 29

Divide using synthetic division. (x⁴−256)/(x−4)

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Identify the divisor and the dividend. Here, the divisor is \(x - 4\), and the dividend is \(x^{4} - 256\).

Set up the synthetic division by writing the zero of the divisor \(x - 4\), which is \(4\), to the left.

Write the coefficients of the dividend polynomial \(x^{4} - 256\). Since some terms are missing, include zeros for those terms: coefficients are \(1\) (for \(x^{4}\)), \(0\) (for \(x^{3}\)), \(0\) (for \(x^{2}\)), \(0\) (for \(x\)), and \(-256\) (constant term).

Perform synthetic division: bring down the first coefficient, multiply it by \(4\), add to the next coefficient, and repeat this process across all coefficients.

Write the quotient polynomial using the results from synthetic division, noting that the degree of the quotient is one less than the dividend, and identify any remainder if present.

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주요 개념

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

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 and performing arithmetic operations in a tabular form.

Polynomial Coefficients

When using synthetic division, it is essential to identify and list the coefficients of the polynomial in descending order of degree. Missing terms must be represented by zero coefficients to maintain the correct alignment during the division process.

Remainder and Quotient Interpretation

The result of synthetic division includes a quotient polynomial and possibly a remainder. The quotient has a degree one less than the original polynomial, and the remainder is a constant. Understanding how to interpret these results is key to completing the division.