Question

factor the expression completely. use the graph to determine the roots for synthetic division. all factors use only integer coefficients and constants.
$x^3 + x^2 - 2$
ways to rewrite expressions
choose an action to perform.
factor common factor
synthetic division

Explanation:

Step1: Identify root from graph

The graph crosses the x-axis at $x=1$, so $(x-1)$ is a factor.

Step2: Set up synthetic division

Use coefficients of $x^3+x^2-2$: $1, 1, 0, -2$ (since there is no $x$ term, coefficient is 0), divide by root $1$.

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

Step3: Write quadratic factor

The result gives quadratic $x^2+2x+2$.

Step4: Check quadratic for factors

Discriminant: $b^2-4ac = 2^2-4(1)(2)=4-8=-4<0$, so it does not factor over integers.

Answer:

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