Simplify each rational expression. Sate any restrictions on the variable. x^2-2x-24/x^2+7x+12 *x^2-1/x-6
Factor the denominators and numerators:
(x-6)(x+4)/(x+4)(x+3) * (x+1)(x-1)/(x-6)
Cancel out factors that are in both the numerator and denominator:
(x-6) / (x+3) * (x+1)(x-1)
Simplify by multiplying the two remaining factors:
(x-6)(x+1)(x-1) / (x+3)
The restrictions on the variable are x ≠ -4 and x ≠ -3.
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Given expression: (x^2-2x-24)/(x^2+7x+12) * (x^2-1)/(x-6)
Step 1: Factor the denominators and numerators
=> (x-6)(x+4)/(x+4)(x+3) * (x+1)(x-1)/(x-6)
=> (x-6)/(x+3) * (x+1)(x-1)
Step 2: Cancel out factors that are in both the numerator and denominator
=> (x-6) / (x+3) * (x+1)(x-1)
Step 3: Simplify by multiplying the two remaining factors
=> (x-6)(x+1)(x-1) / (x+3)
Therefore, the simplified expression is (x-6)(x+1)(x-1) / (x+3). The restrictions on the variable are x ≠ -4 and x ≠ -3 as the expression is undefined at these values of x.
To simplify the given rational expression:
(x^2 - 2x - 24) / (x^2 + 7x + 12) * (x^2 - 1) / (x - 6)
We can factor the numerators and denominators to simplify the expression further.
Factorizing the numerator of the first fraction:
(x^2 - 2x - 24) = (x - 6)(x + 4)
Factorizing the denominator of the first fraction:
(x^2 + 7x + 12) = (x + 3)(x + 4)
So, the first fraction can be expressed as (x - 6)(x + 4) / (x + 3)(x + 4).
Simplifying the second fraction:
(x^2 - 1) = (x - 1)(x + 1).
The second fraction can be written as (x - 1)(x + 1).
Putting the fractions together, we have:
((x - 6)(x + 4) / (x + 3)(x + 4)) * ((x - 1)(x + 1) / (x - 6))
Now, we can simplify the expression by canceling out common factors:
((x - 6)(x + 4) / (x + 3)(x + 4)) * ((x - 1)(x + 1) / (x - 6))
= (x - 1)(x + 1) / (x + 3)
The simplified rational expression is (x - 1)(x + 1) / (x + 3).
As for restrictions on the variable, since we canceled the common factor (x - 6) from both the numerator and denominator, the expression is undefined when x = 6. So, the restriction on the variable is x ≠ 6.
To simplify the rational expression (x^2 - 2x - 24) / (x^2 + 7x +12) * (x^2 - 1) / (x - 6), we can follow these steps:
Step 1: Factor each numerator and denominator.
(x^2 - 2x - 24) factors to (x - 6)(x + 4).
(x^2 + 7x + 12) factors to (x + 3)(x + 4).
(x^2 - 1) factors to (x + 1)(x - 1).
So the expression becomes [(x - 6)(x + 4) / (x + 3)(x + 4)] * [(x + 1)(x - 1) / (x - 6)].
Step 2: Cancel out common factors between the numerators and denominators.
Both (x - 6) and (x + 4) appear in both the numerator and denominator of the first fraction, so they can be canceled out. Similarly, the (x - 6) term can be canceled out in the second fraction.
After canceling the common factors, the expression simplifies to:
[(x + 1)(x - 1)] / (x + 3)
Step 3: State any restrictions on the variable.
To find restrictions on the variable, we need to look for values that would cause a denominator to be equal to zero. In this case, x = -4 and x = -3 are the values that would make the denominator (x + 3) equal to zero.
So, the restrictions on the variable are x ≠ -4 and x ≠ -3.
Therefore, the simplified rational expression is [(x + 1)(x - 1)] / (x + 3), and the restrictions on the variable are x ≠ -4 and x ≠ -3.