Rewrite the expression 2x2+2x−x2+1\frac{2x^2+2x}{-x^2+1} into an equivalent expression having a denominator of x−1x-1.

− 2 x x − 1 -\frac{2x}{x-1} − x − 1 2 x

Rewrite the expression 2 x 2 + 2 x − x 2 + 1 \frac{2x^2+2x}{-x^2+1} into an equivalent expression having a denominator of x − 1 x-1 .

This problem asks us to rewrite the expression two x squared plus two x over negative x squared plus one as an expression having a denominator of x minus one. Now whenever you see these expressions and we're trying to get a certain denominator, a good step to start with is to factor everything out. This oftentimes makes the problem a lot more clear as you're going to solve it. So notice I have a two x in both of these terms in the numerator. So I'm gonna pull that two x to the outside of the parenthesis, and then that's gonna leave us with just an x and then just a one since we're dividing that two x out completely. Then in the denominator, I'm going to factor this piece. Now this is where things get a little bit tricky, because notice that we have a negative x squared plus one. But what I can do is make this whole thing negative, this x squared plus one negative. If I factor a negative out though, I need to change the signs on each of these terms, giving me a positive x squared and a negative one. Now there's a reason that I wrote it like this in the denominator. Because notice that x squared minus one is the same thing as x squared minus one squared. One squared is the same thing as one, so I could just put a square on top of that. But the reason why I'm making this squared is because notice we now have a difference in squares. So in the numerator, I'm gonna keep this two x times x plus one, but in the denominator, well, we'll keep this negative sign. But this whole thing is going to become x plus one times x minus one. That's the difference of squares. Now I can go ahead and bring this negative sign just onto this front factor here, this front term. That's perfectly okay to do because it's still gonna make the whole thing negative. So if two x times x plus one on top, and then and actually, rather than putting it on here, I'm actually going to put it on the second term, because I actually wanna get this x plus one by itself. You'll see here in a minute. So this is gonna be divided by x minus one, and then it's gonna be times or negative x minus one times x plus one. I can switch the order here of any of these because, ultimately, these are all just being multiplied. This is just a negative one that I would be multiplying in here. So it's okay to switch around the order like I just did. But I did this so we could clearly see the x plus one, which is going to cancel. So that's going to give me two x over negative x minus one. Now what I can do at this point is I can go ahead and take this negative sign, and I can just bring it out front of the entire rational expression, giving me negative two x over x minus one. And doing that gives me this denominator I'm looking for of x minus one. So negative two x over x minus one would be the solution to this problem.

Rewrite the expression 2 x 2 + 2 x − x 2 + 1 \frac{2x^2+2x}{-x^2+1} into an equivalent expression having a denominator of x − 1 x-1 .

− 2 x x − 1 -\frac{2x}{x-1} − x − 1 2 x

2 x x − 1 \frac{2x}{x-1} x − 1 2 x

2 x 2 x − 1 \frac{2x^2}{x-1} x − 1 2 x 2

− 2 x 2 x − 1 -\frac{2x^2}{x-1} − x − 1 2 x 2

8. Rational Expressions and Functions

Least Common Denominators

Intermediate Algebra

8. Rational Expressions and Functions

Least Common Denominators

Struggling with Intermediate Algebra?

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Multiple Choice

Rewrite the expression $\frac{2 x^{2} + 2 x}{- x^{2} + 1}$ into an equivalent expression having a denominator of $x minus 1$ .

$\frac{2x}{x-1}$

$-\frac{2x}{x-1}$

$\frac{2x^2}{x-1}$

$-\frac{2x^2}{x-1}$

Verified step by step guidance

Start with the original expression: $\frac{2x^2 + 2x}{-x^2 + 1}$.

Recognize that the denominator $-x^2 + 1$ can be rewritten by factoring out a negative sign: $-x^2 + 1 = -(x^2 - 1)$.

Notice that $x^2 - 1$ is a difference of squares, which factors as $(x - 1)(x + 1)$, so the denominator becomes $-(x - 1)(x + 1)$.

Rewrite the entire fraction using this factorization: $\frac{2x^2 + 2x}{-(x - 1)(x + 1)}$.

To get a denominator of $x - 1$, multiply numerator and denominator by the missing factor to cancel out $(x + 1)$ in the denominator, and simplify the numerator accordingly, keeping track of the negative sign.

6:33m

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