Question

factor the expression completely. use the graph to determine the roots for synthetic division. all factors use only integer coefficients and constants. factor common factor 12x⁴ + 63x³ - 255x² - 756x

Explanation:

Step1: Factor out greatest common factor

Identify the GCF of all terms: $3x$.
$12x^4 + 63x^3 - 255x^2 - 756x = 3x(4x^3 + 21x^2 - 85x - 252)$

Step2: Find rational root for cubic

Use Rational Root Theorem: possible roots are $\pm1, \pm2, \pm3, \pm4, \pm6, \pm7, \pm8, \pm9, \pm12, \pm14, \pm18, \pm21, \pm24, \pm28, \pm36, \pm42, \pm63, \pm72, \pm84, \pm126, \pm252, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{7}{2}, \pm\frac{9}{2}, \pm\frac{21}{2}, \pm\frac{63}{2}, \pm\frac{1}{4}, \pm\frac{3}{4}, \pm\frac{7}{4}, \pm\frac{9}{4}, \pm\frac{21}{4}, \pm\frac{63}{4}$. Test $x=4$:
$4(4)^3 +21(4)^2 -85(4)-252=256+336-340-252=0$. So $x=4$ is a root, factor is $(x-4)$.

Step3: Synthetic division on cubic

Divide $4x^3 +21x^2 -85x -252$ by $(x-4)$:

$$\begin{array}{r|rrrr} 4 & 4 & 21 & -85 & -252 \\ & & 16 & 148 & 252 \\ \hline & 4 & 37 & 63 & 0 \\ \end{array}$$

Result: $4x^2 +37x +63$

Step4: Factor quadratic trinomial

Find two integers that multiply to $4\times63=252$ and add to 37: 9 and 28.
$4x^2 +37x +63 = 4x^2 +28x +9x +63 = 4x(x+7)+9(x+7)=(4x+9)(x+7)$

Step5: Combine all factors

Assemble the complete factored form.

Answer:

$3x(x-4)(4x+9)(x+7)$