Divide using synthetic division. (4x³−3x²+3x−1)÷(x−1)

Verified step by step guidance

1. Identify the divisor and the dividend. Here, the dividend is the polynomial \$4x^{3} - 3x^{2} + 3x - 1\( and the divisor is \)x - 1$.
2. Set up synthetic division by writing the coefficients of the dividend in order: \$4\(, \)-3\(, \)3\(, and \)-1\(. Since the divisor is \)x - 1\(, use \)1\( as the synthetic divisor (the zero of \)x - 1$).
3. Draw the synthetic division setup: write the coefficients in a row and place the divisor root \$1$ to the left.
4. Begin the synthetic division process: bring down the first coefficient \$4\( as is. Then multiply it by \)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.
5. After completing the synthetic division, interpret the bottom row as the coefficients of the quotient polynomial, which will be one degree less than the original polynomial, and the last number as the remainder.

Key Concepts

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 used in the division process to find the quotient and remainder.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This concept helps verify the result of synthetic division and understand the relationship between division and function evaluation.