Q 54E - Intermediate Algebra

Question

In the following exercises, factor.

$x^3+ x^2- x - 1$.

Step-by-Step Solution

Answer

The factors for the expression $x^3+ x^2- x - 1$ are,$\left(x^2-1\right)\left(x+1\right)$.

Step 1 Given expression is,

$x^3+ x^2- x - 1$.

The objective is, we need to find the factor for this expression.

Step 2 First group each term with the common factors.

$x^3+ x^2- x - 1=\left(x^3+ x^2\right)+\left(- x - 1\right)$

Now factor out the common factors.

$$\begin{gathered} x^3+ x^2- x - 1=x^2\left(x+ 1\right)-1\left(x+1\right) \\ =\left(x+1\right)\left(x^2-1\right) \end{gathered}$$

The obtained factors are,$\left(x+1\right)\left(x^2-1\right)$.

Step 3 let us check the answer by multiplying the factors.

$\left(x^2-1\right)\left(x+1\right)=x^3+x^2-x-1$

which is true.