EDGE Math 8

Using a Combination of Methods

Example 3

The volume of a box is given by the polynomial $V = 3 x^{3} + 12 x^{2} + 12 x,$ where x is in centimeters.
a. Express the volume of the box in factored form.
b. Using the polynomial in (a), determine the volume of a box when $x = 5.$

Solution

a. Factor $3 x^{3} + 12 x^{2} + 12 x$ as follows.

3 x^{3} + 12 x^{2} + 12 x

= 3 x (x^{2} + 4 x + 4)

Factor out the GCF.

= 3 x {(x + 2)}^{2}

Factor the perfect square trinomial.

Thus,

3 x^{3} + 12 x^{2} + 12 x = 3 x {(x + 2)}^{2} .

b. The volume of the box when $x = 5$ is

V = 3 x {(x + 2)}^{2}

= 3 (5) {(5 + 2)}^{2}

= 15 {(7)}^{2}

= 15 (49)

= 735

Thus, the volume of the box when $x = 5$ is $735 {cm}^{3} .$