Question

factor the expression completely. use the graph to determine the roots for synthetic division. all factors use only integer coefficients and constants.
$64x^{4}-128x^{3}-1008x^{2}+720x+2592$
synthetic division :enter value:
$16(4x^{4}-8x^{3}-63x^{2}+45x+162)$
factor out a root:

Explanation:

Step1: Identify integer root from graph

The graph crosses the x-axis at $x=1$, $x=2$, $x=-3$, $x=\frac{3}{2}$. We start with integer root $x=1$.

Step2: Factor out GCF first

Factor 16 from $64x^4 - 128x^3 - 1008x^2 + 720x + 2592$:
$16(4x^4 - 8x^3 - 63x^2 + 45x + 162)$

Step3: Synthetic division with $x=1$

For polynomial $4x^4 - 8x^3 - 63x^2 + 45x + 162$, use root $1$:

$$\begin{array}{r|rrrrr} 1 & 4 & -8 & -63 & 45 & 162 \\ & & 4 & -4 & -67 & -22 \\ \hline & 4 & -4 & -67 & -22 & 140 \end{array}$$

Remainder ≠0, try $x=2$:

$$\begin{array}{r|rrrrr} 2 & 4 & -8 & -63 & 45 & 162 \\ & & 8 & 0 & -126 & -162 \\ \hline & 4 & 0 & -63 & -81 & 0 \end{array}$$

Remainder=0, so $x-2$ is a factor. Resulting polynomial: $4x^3 + 0x^2 -63x -81$

Step4: Synthetic division on cubic with $x=3$

For $4x^3 -63x -81$, use root $3$:

$$\begin{array}{r|rrrr} 3 & 4 & 0 & -63 & -81 \\ & & 12 & 36 & -81 \\ \hline & 4 & 12 & -27 & 0 \end{array}$$

Remainder=0, so $x-3$ is a factor. Resulting quadratic: $4x^2 +12x -27$

Step5: Factor the quadratic

Factor $4x^2 +12x -27$:
Find two numbers: $18$ and $-6$, since $18\times(-6)=-162$ and $18-6=12$.
$4x^2 +18x -6x -27 = 2x(2x+9)-3(2x+9)=(2x-3)(2x+9)$

Step6: Rewrite with all factors

Recall the factored out 16 and roots $x=-3$ (from graph, verified as root of original polynomial):

Answer:

$16(x-2)(x-3)(2x-3)(2x+9)$

For the "Factor out a root" prompt, the integer root to enter is $\boldsymbol{2}$ (or $\boldsymbol{3}$, $\boldsymbol{-3}$; $2$ is the first valid integer root found via synthetic division here)