Multiply. State any restrictions on the variable. Show your work.
4x^2+27x+18/16x^2-9 * 4x^2+9x-9/x^2+3x-18
To multiply the given expressions:
(4x^2 + 27x + 18) / (16x^2 - 9) * (4x^2 + 9x - 9) / (x^2 + 3x - 18)
First, factorize the numerator and denominator of each expression:
(4x^2 + 27x + 18) = (4x + 3)(x + 6)
(16x^2 - 9) = (4x + 3)(4x - 3)
(4x^2 + 9x - 9) = (4x - 3)(x + 3)
(x^2 + 3x - 18) = (x + 6)(x - 3)
Now we can cancel out common factors from the numerator and denominator:
((4x + 3)(x + 6) / (4x + 3)(4x - 3)) * ((4x - 3)(x + 3) / (x + 6)(x - 3))
Next, we can simplify further by canceling out the common factors:
(1 / (4x - 3)) * (1 / (x - 3)) = 1 / ((4x - 3)(x - 3))
Therefore, the simplified expression is 1 / ((4x - 3)(x - 3)).
Restrictions on the variable:
The variable x should not equal either 3 or 3/4, as those values would make the denominator zero, resulting in undefined values.
To multiply the given expression:
(4x^2 + 27x + 18) / (16x^2 - 9) * (4x^2 + 9x - 9) / (x^2 + 3x - 18)
First, let's factor the denominators:
16x^2 - 9 = (4x + 3)(4x - 3)
x^2 + 3x - 18 = (x + 6)(x - 3)
Now, let's simplify the expression:
(4x^2 + 27x + 18) * (4x^2 + 9x - 9) / (16x^2 - 9) * (1 / (x^2 + 3x - 18))
Taking out the common factors:
[(4x + 3)(x + 6)] * [ (4x^2 + 9x - 9) / (4x - 3)(x + 6) ] * [ 1 / (x + 6)(x - 3) ]
Notice that the (x + 6) terms cancel out in the numerator and denominator, as well as the (4x - 3) terms in the numerator and denominator:
(4x + 3) * (4x^2 + 9x - 9) / (x - 3)
Now, we multiply using the distributive property:
4x(4x^2 + 9x - 9) + 3(4x^2 + 9x - 9) / (x - 3)
= (16x^3 + 36x^2 - 36x) + (12x^2 + 27x - 27) / (x - 3)
Combining like terms:
= 16x^3 + 36x^2 + 12x^2 + 27x - 36x - 27 / (x - 3)
= 16x^3 + 48x^2 - 9 / (x - 3)
The result of multiplying the given expression is 16x^3 + 48x^2 - 9. There are no restrictions on the variable x.
To multiply the expressions (4x^2+27x+18)/(16x^2-9) and (4x^2+9x-9)/(x^2+3x-18), we can follow these steps:
Step 1: Factorize the denominators.
The first denominator, 16x^2-9, is a difference of squares and can be factored as (4x+3)(4x-3).
The second denominator, x^2+3x-18, can be factored as (x-3)(x+6).
Step 2: Find the common factors.
The common factors between the numerators and denominators are (4x-3) and (x-3).
Step 3: Cancel out the common factors.
By canceling the common factors, we simplify the expression to:
[(4x+6)(x+2)] / [(4x+3)(x+6)]
Step 4: Simplify further, if possible.
In this case, there are no more common factors to cancel. Thus, the expression is simplified as:
(4x+6)(x+2) / (4x+3)(x+6)
Please note that there are no restrictions on the variable x in this case.