I need some help with this question I really don't understand it and some help going through this problem would be really appreciated :)
Use the fact that a/b/c/d=a/b / c/d to simplify each rational expression. State any restrictions on the variables.
x^2-1/x^2-9 / x^2+3x-4/x^2+8x+15
actually, as written, the equation is not true, but we'll let that go.
When you divide by fractions, you invert and multiply:
(a/b) / (c/d) = (a/b)(d/c) = ad/bc
So, you have
(x^2-1/x^2-9)(x^2+8x+15/x^2+3x-4)
Now just factor everybody and start cancelling:
(x-1)(x+1)(x+3)(x+5)
-------------------------
(x-3)(x+3)(x+4)(x-1)
(x+1)(x+5)
---------------
(x-3)(x+4)
No values of x which make any of the denominators (in the original expression, or the result) zero are allowed.
on the abcd, b nor c, nor d can be zero
go one step further on the abcd
it also equals ad/bc
(x^2-1)((x^2+8x+15)/(x^2-9)(x^2+3x-4)
now to do some factoring..
(x+1)(x-1)(x+5)(x+3)/(x+3)(x-3)(x+4)(x-1)
do some dividing out..
(x+1)(x+5)/(x-3)(x+4)
note that x cannot be 3, nor -4
I am so very confused :(
To simplify the given rational expression, we can use the fact that a/b/c/d = a/b * d/c.
Let's break down the given expression step by step and simplify it:
Expression: (x^2 - 1)/(x^2 - 9) / (x^2 + 3x - 4)/(x^2 + 8x + 15)
Step 1: Rewrite the expression using the fact a/b/c/d = a/b * d/c:
(x^2 - 1)/(x^2 - 9) * (x^2 + 8x + 15)/(x^2 + 3x - 4)
Step 2: Factorize the numerator and denominator.
For the first fraction, the numerator:
x^2 - 1 can be factored as (x - 1)(x + 1).
The denominator:
x^2 - 9 can be factored as (x - 3)(x + 3).
For the second fraction, both the numerator and denominator:
x^2 + 8x + 15 can be factored as (x + 3)(x + 5).
x^2 + 3x - 4 cannot be factored further.
So, the simplified expression becomes:
[(x - 1)(x + 1)/(x - 3)(x + 3)] * [(x + 3)(x + 5)/(x^2 + 3x - 4)]
Step 3: Simplify further by canceling out common factors.
In the numerator and denominator of the first fraction, we have (x + 3) common. So, we can cancel it:
[(x - 1)(x + 1)/(x - 3)(x + 3)] * [(x + 3)(x + 5)/(x^2 + 3x - 4)]
= [(x - 1)(x + 1)/(x - 3)] * [(x + 5)/(x^2 + 3x - 4)]
Step 4: Simplify the expression further if possible.
We can check if any other factors can be canceled out, but in this case, there are no more common factors to cancel. So, the expression is fully simplified.
The final simplified expression is:
(x - 1)(x + 1)(x + 5)/(x - 3)(x^2 + 3x - 4)
Make sure to check for any restrictions on the variables.
In this case, we need to ensure that denominators are not equal to zero, which would lead to division by zero and be undefined.
Restrictions:
1. x - 3 should not be equal to zero, meaning x ≠ 3.
2. x^2 + 3x - 4 should not be equal to zero. We can factor this equation: (x - 1)(x + 4) = 0. So, x should not be equal to 1 or -4.
Hence, the restrictions on the variable x are x ≠ 3, 1, and -4.