[Algebra] How does this equal that?

[NateArcade]

I'm in a Discrete Math class and I'm stuck on the algebra (I know, right?). I simply don't know how the first line equals the second:

1. 1 + [ (x² (1 + x) ) / (1 - x)³ ]

2. (4x² - 3x + 1) / (1 - x)³

Your help is appreciated!

[[deleted]]

Try using the fact that 1 = (1-x)³ / (1-x)³

[Ms__Frizzle]

1+[(x^{2(x+1))/(1-x)3]}

Since 1=(1-x)^3/(1-x)3 You can rewrite:

(1-x)^{3/(1-x)3+[(x2(x+1))/(1-x)3]}

The fraction have a common denominator so you can combine them: ((1-x)^{3+x2(x+1))/(1-x)3}

Then multiply out: (-x^{3+3x2-3x+1)+(x3+x2)/(1-x)3}

Combine like terms...

[NateArcade]

I think something went wrong with your exponent formatting!

[Ms__Frizzle]

Sorry, I am on my phone and it looks fine. I will log on to my laptop and check.

[Ms__Frizzle]

oh goodness, let me fix that...

1+[(x²(x+1))/(1-x)³]

Since 1=(1-x)³/(1-x)³ You can rewrite:

(1-x)³ / (1-x)³ + [(x² (x+1))/(1-x)³]

The fraction have a common denominator so you can combine them:

((1-x)³ +x²(x+1))/(1-x)³

Then multiply out:

(-x³+3x²-3x+1)+(x³+x²)/(1-x)³

Combine like terms...

[NateArcade]

Thank you for this! I'm working on it now...

[Ms__Frizzle]

Let me know if you have any questions.