In Exercises 17–32, divide using synthetic division. (x²−5x−5x³+x⁴)÷(5+x)

Verified step by step guidance

1. First, rewrite the dividend polynomial in standard form with descending powers of x: $x{x}^4-5x^3+x^2-5x$. This is $x^4\; -\; 5x^3\; +\; x^2\; -\; 5x$.
2. Next, identify the divisor as $5\; +\; x$. For synthetic division, rewrite it in the form $x\; -\; c$. So, $5\; +\; x\; =\; x\; +\; 5\; =\; x\; -\; (-5)$, which means $c\; =\; -5$.
3. Set up the synthetic division by writing the coefficients of the dividend polynomial in order of descending powers of x. The coefficients are: 1 (for $x^4$), -5 (for $x^3$), 1 (for $x^2$), -5 (for $x^1$), and 0 (for the constant term, since it is missing).
4. Perform synthetic division using $c\; =\; -5$: bring down the first coefficient, multiply it by $c$, add to the next coefficient, and repeat this process across all coefficients.
5. The numbers obtained after synthetic division represent the coefficients of the quotient polynomial, starting from one degree less than the original dividend, and the last number is the remainder.

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.