Question

factor the expression completely. use the graph to determine the roots for synthetic division. all factors use only integer coefficients and constants. view solution history synthetic division :enter value: \\(\frac{x^3 + x^2 - 2}{\text{factor out a root:}}\\)

Explanation:

Step1: Test root $x=1$

Substitute $x=1$ into $x^3+x^2-2$:
$1^3 + 1^2 - 2 = 1 + 1 - 2 = 0$
So $x=1$ is a root, meaning $(x-1)$ is a factor.

Step2: Synthetic division with root 1

Set up synthetic division for $x^3+x^2+0x-2$:

$$\begin{array}{r|rrrr} 1 & 1 & 1 & 0 & -2 \\ & & 1 & 2 & 2 \\ \hline & 1 & 2 & 2 & 0 \\ \end{array}$$

The quotient is $x^2 + 2x + 2$.

Step3: Check quadratic for factors

Discriminant of $x^2+2x+2$:
$\Delta = 2^2 - 4(1)(2) = 4 - 8 = -4 < 0$
No real integer factors, so it is irreducible over integers.

Answer:

$(x-1)(x^2+2x+2)$