In Exercises 17–32, divide using synthetic division. (2x^5−3x^4+x^3−x^2+2x−1)/(x+2)

Verified step by step guidance

1. Identify the divisor and the dividend. The divisor is \(x + 2\), and the dividend is \(2x^5 - 3x^4 + x^3 - x^2 + 2x - 1\).
2. Set the divisor equal to zero to find the root: \(x + 2 = 0\) gives \(x = -2\). This is the number we will use in synthetic division.
3. Write down the coefficients of the dividend: \([2, -3, 1, -1, 2, -1]\).
4. Perform synthetic division by bringing down the leading coefficient (2) and then multiply it by the root (-2), adding the result to the next coefficient, and repeat this process for each coefficient.
5. Continue the process until you have worked through all coefficients, resulting in a new set of coefficients representing the quotient and a remainder.

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form (x - c). It involves using the coefficients of the polynomial and a specific value derived from the divisor to perform the division process more efficiently 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 is 2x^5 - 3x^4 + x^3 - x^2 + 2x - 1, which is a degree 5 polynomial, meaning the highest exponent of x is 5.

Remainder Theorem

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