Question

How do you simplify `[(x^3 - 1)/(x + 4)][(2x - 7)/(x^2 + 3x + 1)]`?

Answer 1

`(2x^4 - 7x^3 - 2x + 7)/(x^3 + 7x^2 + 13x + 4)`

Explanation:

We multiply the numerators with numerators and denominators with denominators.

First, let's look at the numerators.

We multiply the numerators like this:
`(x^3-1)(2x-7)` and now we need to simplify using the rainbow, FOIL, box method, or whatever way you want to do it.

`x^3 * 2x = 2x^4`

`x^3 * -7 = -7x^3`

`-1 * 2x = -2x`

`-1 * -7 = 7`

So if we put them all together, we will get `2x^4 - 7x^3 - 2x + 7`.

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Now multiply the denominators:
`(x + 4)(x^2 + 3x + 1)`

`x * x^2 = x^3`

`x * 3x = 3x^2`

`x * 1 = x`

`4 * x^2 = 4x^2`

`4 * 3x = 12x`

`4 * 1 = 4`

Again, let's put them all together, and we get `x^3 + 3x^2 + x + 4x^2 + 12x + 4`

We still have to combine the "like terms":
`x^3 + 7x^2 + 13x + 4`

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So our final answer is: `(2x^4 - 7x^3 - 2x + 7)/(x^3 + 7x^2 + 13x + 4)`