Question

How do you simplify `\frac { 2x + y } { - 4x + 3y } \times \frac { 24x ^ { 2} - 22x y + 3y ^ { 2} } { - 12x ^ { 2} - 4x y + y ^ { 2} }`?

Answer 1

This is what I got:

Explanation:

`(2x+y)/(-4x+3y) * (24x^2-22xy+3y^2)/(-12x^2-4xy+y^2)`

First, let's put everything onto one fraction:
`((2x+y)(24x^2-22xy+3y^2))/((-4x+3y)(-12x^2-4xy+y^2))`

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Now let's just look at the numerator: `(2x+y)(24x^2-22xy+3y^2)`
We use the distributive property to multiply all of this stuff out:
`2x * 24x^2 = 48x^3`
`2x * -22xy = -44x^2y`
`2x * 3y^2 = 6xy^2`

`y * 24x^2 = 24x^2y`
`y * -22xy = -22xy^2`
`y * 3y^2 = 3y^3`

Let's combine them all together:
`48x^3 - 44x^2y + 6xy^2 + 24x^2y - 22xy^2 + 3y^3`

As you can see, we have a couple things in common here.
Two of these contain `x^2y -> -44x^2y` and `24x^2y`
`-44x^2y + 24x^2y = -20x^2y`

Also `6xy^2 -22xy^2 = -16xy^2`

Now our simplified numerator is `48x^3 - 20x^2y - 16xy^2 + 3y^3`

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Let's look at the denominator: `(-4x+3y)(-12x^2-4xy+y^2)`

Again we use the distributive property to simplify this:
`-4x * -12x^2 = 48x^3`
`-4x * -4xy = 16x^2y`
`-4x * y^2 = -4xy^2`

`3y * -12x^2 = -36x^2y`
`3y * -4xy = -12xy^2`
`3y * y^2 = 3y^3`

Let's combine them all together:
`48x^3 + 16x^2y - 4xy^2 - 36x^2y - 12xy^2 + 3y^3`

`16x^2y` and `-36x^2y` are in common, so:
`16x^2y - 36x^2y = -20x^2y`

`-4xy^2` and `-12xy^2` are in common, so:
`-4xy^2 - 12xy^2 = -16xy^2`

Now our simplified denominator is `48x^3 -20x^2y - 16xy^2 + 3y^3`

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When we put our numerator and denominator together, we get...
`(48x^3 - 20x^2y - 16xy^2 + 3y^3)/(48x^3 -20x^2y - 16xy^2 + 3y^3)`