Question

solve the equation ( v^3 - 2v^2 - 16v = -32 ). the solutions are ( v = square ), ( v = square ), and ( v = square ).

Explanation:

Step1: Rearrange equation to standard form

$v^3 - 2v^2 - 16v + 32 = 0$

Step2: Group terms for factoring

$(v^3 - 2v^2) + (-16v + 32) = 0$

Step3: Factor out common terms

$v^2(v - 2) - 16(v - 2) = 0$

Step4: Factor out binomial

$(v^2 - 16)(v - 2) = 0$

Step5: Factor difference of squares

$(v - 4)(v + 4)(v - 2) = 0$

Step6: Solve for $v$

Set each factor to 0:
$v - 4 = 0 \implies v=4$
$v + 4 = 0 \implies v=-4$
$v - 2 = 0 \implies v=2$

Answer:

$v=2$, $v=4$, and $v=-4$