Use synthetic division to perform each division. (x5 + 3x4 + 2x3 + 2x2 + 3x+1) / x+2

Ch. 3 - Polynomial and Rational Functions

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Ch. 3 - Polynomial and Rational Functions

문제 13

4장, 문제 13

Use synthetic division to perform each division. (x⁵ + 3x⁴ + 2x³ + 2x² + 3x+1) / x+2

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Identify the divisor and rewrite it in the form \( x - c \). Since the divisor is \( x + 2 \), rewrite it as \( x - (-2) \), so \( c = -2 \).

Write down the coefficients of the dividend polynomial \( x^5 + 3x^4 + 2x^3 + 2x^2 + 3x + 1 \) in order: \( 1, 3, 2, 2, 3, 1 \).

Set up the synthetic division tableau by placing \( c = -2 \) to the left and the coefficients in a row to the right.

Bring down the first coefficient (which is 1) directly below the line. Then multiply this number by \( c = -2 \) and write the result under the next coefficient. Add the column and write the sum below the line. Repeat this multiply-and-add process for all coefficients.

After completing the process, the numbers below the line (except the last one) represent the coefficients of the quotient polynomial, starting from degree 4 down to the constant term. The last number is the remainder.

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주요 개념

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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. This method is efficient for finding quotients and remainders quickly.

Polynomial Coefficients and Setup

To use synthetic division, you must identify and list the coefficients of the dividend polynomial in descending order of degree. If any terms are missing, their coefficients are represented as zero. Proper setup ensures accurate calculations during the synthetic division process.

Interpreting the Divisor in Synthetic Division

In synthetic division, the divisor x + 2 is rewritten as x - (-2), so the value used in the process is -2. Recognizing this transformation is crucial because the synthetic division uses the root of the divisor (the value that makes it zero) to perform the division steps correctly.