Divide using synthetic division. (2x²+x−10)÷(x−2)

Verified step by step guidance

1. Identify the divisor and the dividend. The divisor is \( x - 2 \), so the root to use in synthetic division is \( 2 \). The dividend is \( 2x^{2} + x - 10 \).
2. Write down the coefficients of the dividend in descending order of powers: \( 2 \) (for \( x^{2} \)), \( 1 \) (for \( x \)), and \( -10 \) (constant term).
3. Set up the synthetic division by writing the root \( 2 \) on the left and the coefficients \( 2, 1, -10 \) in a row to the right.
4. Perform synthetic division steps: bring down the first coefficient, multiply it by the root, add the result to the next coefficient, and repeat until all coefficients are processed.
5. Interpret the final row of numbers as the coefficients of the quotient polynomial and the remainder. The degree of the quotient is one less than the dividend's degree.

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 polynomials, making calculations faster and less error-prone.

Polynomial Coefficients

In synthetic division, you work with the coefficients of the dividend polynomial arranged in descending order of degree. Understanding how to identify and organize these coefficients correctly is essential for applying synthetic division accurately.

Division by a Linear Binomial

Synthetic division applies specifically when dividing by a linear binomial like (x - c). Recognizing the divisor's form allows you to set up the synthetic division process by using the value c, which is the root of the divisor polynomial.