Divide using synthetic division. (3x²+7x−20)÷(x+5)

Verified step by step guidance

1. Identify the divisor and rewrite it in the form \( x - c \). Since the divisor is \( x + 5 \), rewrite it as \( x - (-5) \), so \( c = -5 \).
2. Write down the coefficients of the dividend polynomial \( 3x^2 + 7x - 20 \). These are \( 3, 7, \) and \( -20 \).
3. Set up the synthetic division by placing \( c = -5 \) to the left and the coefficients \( 3, 7, -20 \) to the right.
4. Perform synthetic division steps: bring down the first coefficient, multiply it by \( c \), add 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 quotient will have one degree less than the original polynomial.

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 Division

Polynomial division involves dividing one polynomial by another, similar to numerical division. Understanding how to divide polynomials helps in simplifying expressions, finding factors, and solving polynomial equations.

Coefficients and Remainders

In synthetic division, only the coefficients of the dividend polynomial are used, and the remainder is the final value obtained after the division process. Recognizing how to interpret these coefficients and the remainder is essential for writing the quotient and remainder correctly.