You are given an expression for the volume of the rectangular prism. Determine an expression for the missing dimension. V=4x^3+5x^2-32x-33 and the dimensions are (x+1) and (x+3

Answer 1

Hi Callie,

When solving for volume of a rectangular prism, you want to multiply all of the dimensions together in order to get the total volume. Given any two dimensions and volume, we can therefore find the third.

We will call this third dimension y for now

Volume = 4x³+5x²-32x-33 = (x+1)(x+3)(y)

= x²y+4xy+3y

We know that y cannot contain a power of x greater than 1. This is because if y were x², there would be an x⁴, which is too high for this equation. We also know that y cannot contain a power of x less than 1. This is because it would be impossible to receive an x³ in such a situation. As such, y must be equal to (nx+z) for some number n and z. Given that the coefficient in front of x³ is a 4, we can assume that n must also be 4, so y=(4x+z). Given that the only term without an x is equal to -33, and we have 3y, we find that z must be -11, making y=(4x-11).

Let's double check our logic here:

x²(4x-11)+4x(4x-11)+3(4x-11)=4x³-11x²+16x²-44x+12x-33=4x³+5x²-32x-33.

The reason this method works is because, when we multiply out, there is only one of each of the end terms, those being the x³ term and the x⁰ term.

The more traditional method to solve this problem would be by dividing out the two dimensions given, which we know are factors. This can be done via either long division or synthetic division. For the purposes of this example, I will use long division.

(4x³+5x²-32x-33)/(x+1)

x+1 goes into 4x³+5x² a total of 4x² times, with x² left over afterwards

x+1 goes into x²-32x a total of x times, with -33x left over afterwards

x+1 goes into -33x-33 a total of -33 times, with nothing left over.

4x³+5x²-32x-33 = (x+1)(4x²+x-33)

Let's do it again with (x+3)

(4x²+x-33)/(x+3)

x+3 goes into 4x²+x a total of 4x times, with -11x left over afterwards

x+3 goes into -11x-33 a total of -11 times, with nothing left over

4x³+5x²-32x-33 = (x+1)(4x²+x-33)=(x+1)(x+3)(4x-11)

If you have any questions, please feel free to reach out or leave a comment below.

Alim Merchant

Answer 2

Use synthetic substitution with the values -1 and -3 to suppress the polynomial to the third linear factor.