In Exercises 17–32, divide using synthetic division. (2x⁵−3x⁴+x³−x²+2x−1)/(x+2)

Verified step by step guidance

1. 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.
2. Write down the coefficients of the dividend polynomial 2x^5 - 3x^4 + x^3 - x^2 + 2x - 1 in order: 2, -3, 1, -1, 2, -1.
3. Set up synthetic division by writing c = -2 to the left and the coefficients in a row to the right.
4. Bring down the first coefficient (2) as it is. Then multiply it by c (-2) 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. The numbers obtained at the bottom row (except the last one) are the coefficients of the quotient polynomial, starting from degree 4 down to the constant term. The last number is 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 manipulated to find the quotient and remainder efficiently.

Divisor in the Form x + c

When dividing by a binomial like x + 2, synthetic division uses the root of the divisor, which is the value that makes the divisor zero (x = -2). This value is used in the synthetic division process to perform the calculations.