Use synthetic division to perform each division. (x3 + 3x2 +11x + 9) / x+1

Ch. 3 - Polynomial and Rational Functions

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Ch. 3 - Polynomial and Rational Functions

문제 7

4장, 문제 7

Use synthetic division to perform each division. (x³ + 3x² +11x + 9) / x+1

검증된 단계별 안내

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 \(x^3 + 3x^2 + 11x + 9\). These are \(1\), \(3\), \(11\), and \(9\).

Set up the synthetic division by placing \(c = -1\) to the left and the coefficients \(1\), \(3\), \(11\), \(9\) in a row to the right.

Bring down the first coefficient (1) as is. Multiply it by \(c\) (which is \(-1\)) 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) represent the coefficients of the quotient polynomial, and the last number is the remainder. Write the quotient polynomial using these coefficients with decreasing powers of \(x\) starting from one degree less than the original polynomial.

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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, making calculations faster and less error-prone.

Polynomial Coefficients

Polynomial coefficients are the numerical factors in front of the variable terms. In synthetic division, these coefficients are arranged in descending order of degree and manipulated to find the quotient and remainder efficiently.

Division by a Linear Binomial

Dividing by a linear binomial like x + 1 involves rewriting it in the form x - (-1) to identify the value used in synthetic division. This step is crucial because synthetic division uses the root of the divisor to perform the calculation.