Question

factor the expression completely. use the graph to determine the roots for synthetic division. all factors use only integer coefficients and constants. view solution history factor common factor $16x^3 + 116x^2 + 188x + 40$ use the optional scratchpad. enter the result.

Explanation:

Step1: Factor out GCF

Find the greatest common factor (GCF) of 16, 116, 188, 40. The GCF is 4.
$16x^3 + 116x^2 + 188x + 40 = 4(4x^3 + 29x^2 + 47x + 10)$

Step2: Identify rational root

Use Rational Root Theorem: possible roots are $\pm1, \pm2, \pm5, \pm10, \pm\frac{1}{2}, \pm\frac{5}{2}, \pm\frac{1}{4}, \pm\frac{5}{4}$. Test $x=-2$:
$4(-2)^3 + 29(-2)^2 + 47(-2) + 10 = -32 + 116 - 94 + 10 = 0$. So $x=-2$ is a root, factor is $(x+2)$.

Step3: Synthetic division

Divide $4x^3 + 29x^2 + 47x + 10$ by $(x+2)$ using synthetic division:

-2 | 4   29   47   10
     |     -8  -42  -10
     ----------------
       4   21    5    0

Result is $4x^2 + 21x + 5$.

Step4: Factor quadratic

Factor $4x^2 + 21x + 5$: find two numbers that multiply to $4*5=20$ and add to 21 (20 and 1).
$4x^2 + 20x + x + 5 = 4x(x+5) + 1(x+5) = (4x+1)(x+5)$

Step5: Combine all factors

Multiply the GCF and the linear factors.
$4(x+2)(4x+1)(x+5)$

Answer:

$4(x+2)(x+5)(4x+1)$