Simplifying expressions

[kimd6746]

How do you simplify the following equation?

(3x^2(x-2) - x + 2) / x - 2 = ?

Why am I not allowed to cancel the denominator by multiplying both sides by x-2? Even if I don't cancel out the denominator and try to combine the numerator, I get:

3x^3 - 6x^2 -x + 2 / x - 2

The trail stops cold there. What am I missing?

[Guest]

How do you simplify the following equation?

(3x^2(x-2) - x + 2) / x - 2 = ?

Why am I not allowed to cancel the denominator by multiplying both sides by x-2? Even if I don't cancel out the denominator and try to combine the numerator, I get:

3x^3 - 6x^2 -x + 2 / x - 2

The trail stops cold there. What am I missing?

You can't cancel out x-2 because it's part of the expression (3x^2(x-2) - x + 2). You can only cancel out if x-2 was a multiplier of the full expression.

[rfernandez]

Guest makes a good point. What's not immediately apparent, though, is that (x-2) *is* a factor of the numerator. You just have to massage it a little:

[3(x^2)*(x-2) - x + 2] / (x - 2) (I changed the notation a little for clarity)
[3(x^2)*(x-2) - (x - 2)] / (x - 2)
[(x - 2) [3(x^2) - 1] / (x - 2)
This is the tricky step. factor out (x-2) from both terms of the numerator, leaving behind two terms: 3(x^2) and -1, or [3(x^2) - 1]. Now, there's a (x - 2) factor in the numerator and the denominator, so cancel.

3(x^2) - 1


Rey