Question

write the expression for the volume of the prism. simplify your answer and write it in standard form.

(image of a rectangular prism with length (7x - 8), width (2x), and height (3x + 4))

Explanation:

Step1: Recall the volume formula for a rectangular prism

The volume \( V \) of a rectangular prism is given by the product of its length, width, and height. So, \( V = l \times w \times h \), where \( l = 7x - 8 \), \( w = 2x \), and \( h = 3x + 4 \).

Step2: Multiply the first two factors

First, multiply \( (7x - 8) \) and \( 2x \). Using the distributive property (also known as the distributive law of multiplication over addition/subtraction), we get:
\( (7x - 8) \times 2x = 7x \times 2x - 8 \times 2x = 14x^2 - 16x \)

Step3: Multiply the result by the third factor

Now, multiply \( (14x^2 - 16x) \) by \( (3x + 4) \). Using the distributive property (FOIL method extended for polynomials), we have:
\( (14x^2 - 16x)(3x + 4) = 14x^2 \times 3x + 14x^2 \times 4 - 16x \times 3x - 16x \times 4 \)
Calculate each term:

• \( 14x^2 \times 3x = 42x^3 \)
• \( 14x^2 \times 4 = 56x^2 \)
• \( -16x \times 3x = -48x^2 \)
• \( -16x \times 4 = -64x \)

Step4: Combine like terms

Combine the \( x^2 \) terms: \( 56x^2 - 48x^2 = 8x^2 \)
So, putting it all together, we get:
\( 42x^3 + 8x^2 - 64x \)

Answer:

\( 42x^3 + 8x^2 - 64x \)