Questions: Use synthetic division to find the result when 3x^3-4x^2-1 is divided by x-1. If there is a remainder, express the result in the form q(x)+r(x)/b(x).

Transcript text: Use synthetic division to find the result when $3 x^{3}-4 x^{2}-1$ is divided by $x-1$. If there is a remainder, express the result in the form $q(x)+\frac{r(x)}{b(x)}$.

Solution

Solution Steps

To solve the problem using synthetic division, follow these steps:

Identify the coefficients of the polynomial $3x^3 - 4x^2 - 1$.
Set up the synthetic division using the root of the divisor $x - 1$, which is $1$.
Perform the synthetic division process to find the quotient and remainder.
Express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Step 1: Identify the Polynomial and Divisor

We are given the polynomial $3x^3 - 4x^2 - 1$ and we need to divide it by the divisor $x - 1$.

Step 2: Perform Synthetic Division

Using synthetic division with the root $1$ (from $x - 1 = 0$), we find the coefficients of the polynomial:

Coefficients: $3, -4, 0, -1$

The synthetic division process yields the following results:

Quotient: $3x^2 - x - 1$
Remainder: $-2$

Step 3: Express the Result

The result of the division can be expressed in the form: \[ q(x) + \frac{r(x)}{b(x)} \] where $q(x) = 3x^2 - x - 1$, $r(x) = -2$, and $b(x) = x - 1$. Thus, we can write: \[ 3x^2 - x - 1 + \frac{-2}{x - 1} \]