Rewrite the expression 7x2+7xx2−1\frac{7x^2+7x}{x^2-1} into an equivalent expression having a denominator of x−1x-1

7 x x − 1 \frac{7x}{x-1} x − 1 7 x

Rewrite the expression 7 x 2 + 7 x x 2 − 1 \frac{7x^2+7x}{x^2-1} into an equivalent expression having a denominator of x − 1 x-1

this problem, we are asked to rewrite the expression seven x squared plus seven x over x squared minus one into an equivalent expression having a denominator of x minus one. So we're trying to get a denominator of just x minus one from what we have right about here. So we have seven x squared plus seven x, and then in the denominator, we have x squared minus one. Now it's kinda tricky to see how to do this if we're just taking a look directly at this expression because everything is not factored at the moment. Usually, a straightforward way of dealing with any of these rational expressions is to factor first. So what I'm gonna start by doing is factoring the top. I can take a seven x in common out of each of these terms. That'll give me seven x on the outside of these parentheses. And dividing each of these terms by seven x will leave me with just an x there, then we'll have plus seven x over seven x will just be one. So I'm dividing that seven x out, bringing it to the outside of the parentheses. And then in the denominator, we have x squared minus one. X squared minus one is the same thing as x squared minus one squared. One squared is just equal to one. Now why would I write it like this? Well, notice this gives us a difference of squares. So x squared minus one squared can be factored into x plus one times x minus one. And notice I can just cancel the x plus one. So what that's gonna leave me with is seven x in the numerator, and then in the denominator, we'll have x minus one. Notice x minus one is the denominator we want it to have, so this is the solution to the problem.

Rewrite the expression 7 x 2 + 7 x x 2 − 1 \frac{7x^2+7x}{x^2-1} into an equivalent expression having a denominator of x − 1 x-1

7 x x − 1 \frac{7x}{x-1} x − 1 7 x

7 x 2 x − 1 \frac{7x^2}{x-1} x − 1 7 x 2

7 x 2 + 7 x − 1 \frac{7x^2+7}{x-1} x − 1 7 x 2 + 7

7 x 2 − 7 x − 1 \frac{7x^2-7}{x-1} x − 1 7 x 2 − 7

8. Rational Expressions and Equations

Least Common Denominators

Beginning & Intermediate Algebra

8. Rational Expressions and Equations

Least Common Denominators

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Multiple Choice

Rewrite the expression $\frac{7 x^{2} + 7 x}{x^{2} - 1}$ into an equivalent expression having a denominator of $x minus 1$

$\frac{7x^2}{x-1}$

$\frac{7x^2+7}{x-1}$

$\frac{7x}{x-1}$

$\frac{7x^2-7}{x-1}$

Verified step by step guidance

Start with the original expression: $\frac{7x^2 + 7x}{x^2 - 1}$.

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

Rewrite the expression using the factored denominator: $\frac{7x^2 + 7x}{(x - 1)(x + 1)}$.

Factor the numerator by taking out the common factor \$7x$: $7x^2 + 7x = 7x(x + 1)$, so the expression becomes $\frac{7x(x + 1)}{(x - 1)(x + 1)}$.

Cancel the common factor $(x + 1)$ from numerator and denominator to get an equivalent expression with denominator $x - 1$: $\frac{7x}{x - 1}$.

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