Use synthetic division to perform each division. (5x4 +5x3 + 2x2 - x-3) / x+1

Ch. 3 - Polynomial and Rational Functions

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Ch. 3 - Polynomial and Rational Functions

문제 9

4장, 문제 9

Use synthetic division to perform each division. (5x⁴ +5x³ + 2x² - x-3) / x+1

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Identify the divisor and rewrite it in the form \( x - c \). Since the divisor is \( x + 1 \), rewrite it as \( x - (-1) \), so \( c = -1 \).

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

Set up the synthetic division by placing \( c = -1 \) to the left and the coefficients to the right in a row.

Bring down the first coefficient (5) as it is. Then multiply it by \( c = -1 \) 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 at the bottom row (except the last one) are the coefficients of the quotient polynomial, starting from one degree less than the original polynomial. 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 first 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

When dividing by a binomial like x + 1, rewrite it in the form x - c to identify the value of c used in synthetic division. For x + 1, c is -1. This value is used in the synthetic division process to multiply and combine coefficients correctly to find the quotient and remainder.