Question

1. The volume of a rectangular prism (shown below) is 8x4 +20x³ +8x². What is one dimension of the prism?

2x²
length
O (x-2)
O (2x+1)
O (2x+2)
height

Answer 1

(2x + 1)

Explanation:

the volume (V) of a rectangular prism is calculated as

V = length × width × height

given

V = 8
`x^(4)` + 20x³ + 8x² ← factor out a common factor of 4x² from each term

= 4x²(2x² + 5x + 2) ← factor the quadratic expression

consider the product of the factors of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term.

product = 2 × 2 = 4 and sum = + 5

the factors are + 4 and + 1

use these factors to split the x- term

2x² + 4x + x + 2 ( factor the first/second and third/fourth terms )

= 2x(x + 2) + 1(x + 2) ← factor out (x + 2) from each term

= (x + 2)(2x + 1)

thus

V = 4x²(x + 2)(2x + 1)

then (2x + 1) is one dimension of the prism