In Exercises 17–32, divide using synthetic division. (3x^2+7x−20)÷(x+5)

Verified step by step guidance

1. Identify the divisor and the dividend. The divisor is $x+5$ and the dividend is $3x^2+7x-20$.
2. Set the divisor equal to zero to find the root: $x+5=0$, so $x=-5$. This is the number we will use in synthetic division.
3. Write down the coefficients of the dividend: $3,7,-20$.
4. Perform synthetic division: Bring down the first coefficient (3) as it is. Multiply it by the root (-5) and add to the next coefficient (7). Repeat this process for the remaining coefficients.
5. The result will give you the coefficients of the quotient polynomial and a remainder, if any.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing polynomials, particularly useful when dividing by linear factors. It involves using the coefficients of the polynomial and a specific value derived from the divisor to perform the division in a streamlined manner, avoiding the more complex long division process.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, the polynomial 3x^2 + 7x - 20 is a quadratic function, which is essential to understand as it represents the dividend in the division process.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is equal to f(c). This theorem is useful in synthetic division as it helps to quickly determine the remainder without performing the entire division process.