Multiply. State any restrictions on the variable. Show your work.
2x^2+13x+15/4x^2-9 * 2x^2+3x-9/x^2+2x-15
To multiply these expressions, we first factor each of the quadratic expressions:
2x^2 + 13x + 15 = (2x + 3)(x + 5)
4x^2 - 9 = (2x + 3)(2x - 3)
2x^2 + 3x - 9 = (2x - 3)(x + 3)
x^2 + 2x - 15 = (x + 5)(x - 3)
Now, we can cancel out common factors in the numerator and denominator:
(2x + 3)(x + 5)/(2x + 3)(2x - 3) * (2x - 3)(x + 3)/(x + 5)(x - 3)
After canceling out common factors, we are left with:
(x + 3)/(x - 3)
There are no restrictions on the variable x.
To multiply the given expressions:
(2x^2 + 13x + 15) / (4x^2 - 9) * (2x^2 + 3x - 9) / (x^2 + 2x - 15)
Let's simplify the expressions first:
Factor the denominators:
4x^2 - 9 = (2x + 3)(2x - 3)
x^2 + 2x - 15 = (x + 5)(x - 3)
Now we have:
(2x^2 + 13x + 15) / ((2x + 3)(2x - 3)) * (2x^2 + 3x - 9) / ((x + 5)(x - 3))
Next, simplify the numerators:
2x^2 + 13x + 15 = (2x + 3)(x + 5)
2x^2 + 3x - 9 = (2x - 3)(x + 3)
Now we have:
[(2x + 3)(x + 5) / ((2x + 3)(2x - 3))] * [(2x - 3)(x + 3) / ((x + 5)(x - 3))]
Cancel out common factors:
[(2x + 3)(cancel(x + 5)) / ((2x + 3)(2x - 3))] * [(2x - 3)(x + 3) / (cancel(x + 5)(x - 3))]
Simplifying further:
(2x + 3) / (2x - 3) * (2x - 3)(x + 3)
Cancelling out the (2x - 3) terms:
(2x + 3) * (x + 3)
Expanding:
2x^2 + 6x + 3x + 9
Combining like terms:
2x^2 + 9x + 9
Now, we have the final expression after simplification:
2x^2 + 9x + 9
There are no restrictions on the variable in this multiplication problem.
To simplify the given expression and state any restrictions on the variable, we need to follow these steps:
Step 1: Factor the polynomials in both the numerator and the denominator.
First, factor the numerator: 2x^2 + 13x + 15.
This can be factored as: (2x + 3)(x + 5).
Next, factor the denominator: 4x^2 - 9.
This is a difference of squares and can be factored as: (2x + 3)(2x - 3).
Then, factor the second numerator: 2x^2 + 3x - 9.
Unfortunately, this cannot be factored further using simple integer factors, so we will leave it as is.
Finally, factor the second denominator: x^2 + 2x - 15.
This can be factored as: (x + 5)(x - 3).
Step 2: Cancel out any common factors between the numerator and denominator.
In this case, we see that (2x + 3) and (x + 5) appear in both the numerator and denominator. They can be canceled out, leaving:
(2x - 3) * (2x^2 + 3x - 9) / (x - 3) * (x + 5).
Step 3: Simplify the expression further if possible.
The expression is now in its simplest factored form, and it cannot be simplified any further.
Step 4: State any restrictions on the variable.
To find restrictions on the variable, we need to look for any values that would make the denominator equal to zero. In this case, we see that (x - 3) and (x + 5) are both factors in the denominator, so we need to make sure that x ≠ 3 and x ≠ -5. These values would make the denominator zero, which is not allowed.
Therefore, the restrictions on the variable are x ≠ 3 and x ≠ -5.
To summarize:
The simplified expression, factoring the numerator and denominator, is:
(2x - 3) * (2x^2 + 3x - 9) / (x - 3) * (x + 5).
The restrictions on the variable are: x ≠ 3 and x ≠ -5.