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

Verified step by step guidance

1. Rearrange the polynomial in descending order of powers: $x^4-x^3-5x^2+0x+0$.
2. Identify the divisor $(5+x)$ and rewrite it in the form $(x-(-5))$ to use synthetic division.
3. Set up the synthetic division by writing $-5$ (the opposite of the constant term in the divisor) and the coefficients of the dividend $[1,-1,-5,0,0]$.
4. Bring down the leading coefficient $1$ to the bottom row.
5. Multiply $-5$ by the number just written on the bottom row, add this result to the next coefficient, and continue this process across all coefficients.

Key Concepts

Synthetic Division

Synthetic division is a simplified method of 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 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. Understanding the structure of polynomial functions, including their degree and leading coefficient, is essential for performing operations like division and identifying their behavior.

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.