In Exercises 17–32, divide using synthetic division. (2x^2+x−10)÷(x−2)

Verified step by step guidance

1. Identify the divisor and the dividend. The divisor is \(x - 2\) and the dividend is \(2x^2 + x - 10\).
2. Set up synthetic division by writing the zero of the divisor, which is \(x = 2\), on the left side.
3. Write the coefficients of the dividend \(2x^2 + x - 10\) in order: \(2, 1, -10\).
4. Bring down the leading coefficient \(2\) to the bottom row.
5. Multiply the number you just brought down (\(2\)) by the zero of the divisor (\(2\)) and write the result under the next coefficient (\(1\)). Add this result to the coefficient above it and write the sum below. Repeat this process for the remaining coefficients.

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 (the root of the divisor) to perform the division in a more efficient manner than traditional long division.

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 (2x^2 + x - 10) is a quadratic function, which is essential to understand when performing operations like division.

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 fully performing the division.