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 \( 2x^{5} - 3x^{4} + x^{3} - x^{2} + 2x - 1 \). The coefficients are: \( 2, -3, 1, -1, 2, -1 \).
3. Set up the 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 one degree less than the original polynomial. 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 and Degree

Understanding the coefficients and degree of a polynomial is essential for synthetic division. The degree indicates the highest power of x, and coefficients are the numerical factors of each term, which are arranged in descending order of degree for the division process.

Division by a Binomial of the Form x + c

When dividing by a binomial like x + 2, it is rewritten as x - (-2) to apply synthetic division. Recognizing this form allows you to correctly identify the value of c (here, -2) used in the synthetic division steps.