Question

factor the expression completely. use the graph to determine the roots for synthetic division. all factors use only integer coefficients and constants.
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factor common factor
$6x^3 - 63x^2 + 729$
use the optional scratchpad.
enter the result.

Explanation:

Step1: Find greatest common factor

Identify the GCF of 6, 63, 729. The GCF is 3.

Step2: Factor out the GCF

Divide each term by 3 and factor it out.
$6x^3 - 63x^2 + 729 = 3(2x^3 - 21x^2 + 243)$

Step3: Check rational roots (Rational Root Theorem)

Possible rational roots are $\pm1, \pm3, \pm9, \pm27, \pm81, \pm243, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{9}{2}, \pm\frac{27}{2}, \pm\frac{81}{2}, \pm\frac{243}{2}$. Test $x=9$:
$2(9)^3 -21(9)^2 +243 = 2(729)-21(81)+243 = 1458-1701+243=0$. So $x=9$ is a root, meaning $(x-9)$ is a factor.

Step4: Synthetic division with root 9

Divide $2x^3 -21x^2 +0x +243$ by $(x-9)$:

$$\begin{array}{r|rrrr} 9 & 2 & -21 & 0 & 243 \\ & & 18 & -27 & -243 \\ \hline & 2 & -3 & -27 & 0 \end{array}$$

Resulting quadratic: $2x^2 -3x -27$

Step5: Factor the quadratic

Factor $2x^2 -3x -27$. Find two numbers: product $2*(-27)=-54$, sum $-3$. The numbers are $-9$ and $6$.
Rewrite and factor by grouping:
$2x^2 +6x -9x -27 = 2x(x+3)-9(x+3)=(2x-9)(x+3)$

Step6: Combine all factors

Multiply the GCF with all linear factors.

Answer:

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