Question

use synthetic division to find the result when $4x^4 - 9x^3 + 14x^2 - 12x - 1$ is divided by $x - 1$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Set up synthetic division

For dividing by \(x - 1\), we use \(c = 1\). The coefficients of the polynomial \(4x^{4}-9x^{3}+14x^{2}-12x - 1\) are \(4\), \(-9\), \(14\), \(-12\), \(-1\).
Set up the synthetic division as:

$$
LATEXBLOCK0
$$

Step2: Interpret the results

The numbers in the bottom row (excluding the last one) are the coefficients of the quotient polynomial \(q(x)\), and the last number is the remainder \(r(x)\).
The quotient polynomial \(q(x)\) has degree \(3\) (since we divided a degree \(4\) polynomial by a degree \(1\) polynomial) with coefficients \(4\), \(-5\), \(9\), \(-3\), so \(q(x)=4x^{3}-5x^{2}+9x - 3\). The remainder \(r(x)=- 4\), and the divisor \(b(x)=x - 1\).
So the result is \(4x^{3}-5x^{2}+9x - 3+\frac{-4}{x - 1}\) or \(4x^{3}-5x^{2}+9x - 3-\frac{4}{x - 1}\).

Answer:

\(4x^{3}-5x^{2}+9x - 3-\frac{4}{x - 1}\)