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
15x³ - 85x² + 60x + 160
use the optional scratchpad.
enter the result.

Explanation:

Step1: Factor out GCF

The greatest common factor of 15, 85, 60, 160 is 5.
$15x^3 - 85x^2 + 60x + 160 = 5(3x^3 - 17x^2 + 12x + 32)$

Step2: Find integer root via Rational Root Theorem

Possible roots: $\pm1, \pm2, \pm4, \pm8, \pm16, \pm32, \pm\frac{1}{3}, \pm\frac{2}{3}, ...$
Test $x=4$: $3(4)^3 -17(4)^2 +12(4)+32=192-272+48+32=0$. So $x=4$ is a root, factor is $(x-4)$.

Step3: Synthetic division on cubic

Divide $3x^3 -17x^2 +12x +32$ by $(x-4)$:

4 | 3  -17   12   32
        12  -20  -32
      3  -5   -8    0

Result: $3x^2 -5x -8$

Step4: Factor quadratic trinomial

Find two numbers: product $3*(-8)=-24$, sum $-5$: $-8$ and $3$.
$3x^2 -5x -8 = 3x^2 +3x -8x -8 = 3x(x+1)-8(x+1)=(3x-8)(x+1)$

Step5: Combine all factors

Multiply the GCF and the linear factors.
$5(x-4)(3x-8)(x+1)$

Answer:

$5(x-4)(3x-8)(x+1)$