Question

Simplify each rational expression. State any restrictions on the variable.
X^2-2x-24/x^2+7x+12 * x^2-1/x-6

Answer 1

wrong. It is (x^2 - 1) / (x + 3)

Answer 2

also, from the original equation, x ≠ -3, -4, 6

Answer 3

To simplify the expression, we factor each polynomial:

(x^2 - 2x - 24) / (x^2 + 7x + 12) * (x^2 - 1) / (x - 6)

Factoring the polynomials, we get:

(x - 6)(x + 4) / (x + 3)(x + 4) * (x + 1)(x - 1) / (x - 6)

Now, we can cancel out the common factors:

((x - 6) * x * (x + 1) * (x - 1)) / ((x + 3) * (x - 6))

The final simplified expression is:

x(x^2 - 1) / (x + 3)

Therefore, the simplified rational expression is x(x^2 - 1) / (x + 3). The restriction on the variable is x ≠ -3 and x ≠ 6.

Answer 4

Apologies for the oversight. Let's simplify the expression again:

(x^2 - 2x - 24) / (x^2 + 7x + 12) * (x^2 - 1) / (x - 6)

Factoring the polynomials in the numerator, we have:

(x - 6)(x + 4) / (x + 3)(x + 4) * (x + 1)(x - 1) / (x - 6)

Now, simplifying and canceling out common factors:
(x - 6)(x + 1)(x - 1) / (x + 3)

Simplifying further:
(x^2 - 1) / (x + 3)

So, the correct simplified expression is (x^2 - 1) / (x + 3). The restriction on the variable is x ≠ -3.

Answer 5

Yes, you are correct. Thank you for pointing that out.

The correct simplified expression is (x^2 - 1) / (x + 3) and the restrictions on the variable are x ≠ -3, -4, 6. Thank you for your patience.