Question

factor the expression completely. use the graph to determine the roots for synthetic division. all factors use only integer coefficients and constants. synthetic division :enter value \\(x^3 - 7x + 36\\) factor out a root:

Explanation:

Step1: Identify root from graph

The graph crosses the x-axis at $x=-4$, so $x=-4$ is a root, meaning $(x+4)$ is a factor.

Step2: Set up synthetic division

Use root $-4$ for polynomial $x^3+0x^2-7x+36$:

$$\begin{array}{r|rrrr} -4 & 1 & 0 & -7 & 36 \\ & & -4 & 16 & -36 \\ \hline & 1 & -4 & 9 & 0 \end{array}$$

Step3: Write quotient polynomial

The quotient is $x^2-4x+9$, which has no real roots (discriminant $\Delta=(-4)^2-4(1)(9)=16-36=-20<0$).

Step4: Combine factors

Multiply the linear factor and quadratic factor.

Answer:

The fully factored form is $(x+4)(x^2-4x+9)$, and the root to factor out first is $\boldsymbol{-4}$