Multiply. Show any restrictions on the variable. Show your work.
4x^2+27x+18/16x^2-9 * 4x^2+9x-9/x^2+3x-18
To simplify the expression, let's factor all the quadratic expressions first:
4x^2 + 27x + 18 = (2x + 3)(2x + 6)
16x^2 - 9 = (4x + 3)(4x - 3)
4x^2 + 9x - 9 = (2x - 1)(2x + 9)
x^2 + 3x - 18 = (x - 3)(x + 6)
Now we can rewrite the expression:
[(2x + 3)(2x + 6) / (4x + 3)(4x - 3)] * [(2x - 1)(2x + 9) / (x - 3)(x + 6)]
Next, we can simplify the expression by canceling out common factors in the numerator and denominator:
[(2x + 3)(2x + 6)(2x - 1)(2x + 9)] / [(4x + 3)(4x - 3)(x - 3)(x + 6)]
Now that no common factors remain, we can multiply the remaining factors:
(2x + 3)(2x + 6)(2x - 1)(2x + 9) / (4x + 3)(4x - 3)(x - 3)(x + 6)
So the simplified expression is:
(2x + 3)(2x + 6)(2x - 1)(2x + 9) / (4x + 3)(4x - 3)(x - 3)(x + 6)
There are no restrictions on the variable x in this case.
To multiply the given expressions, we first need to factorize both of them.
Let's factorize the numerator of the first expression: 4x^2 + 27x + 18.
The quadratic expression can be factored as (4x + 3)(x + 6).
Next, let's factorize the denominator of the first expression: 16x^2 - 9.
This is a difference of squares, which can be factored as (4x + 3)(4x - 3).
Now, let's factorize the numerator of the second expression: 4x^2 + 9x - 9.
The quadratic expression cannot be factored any further.
Lastly, let's factorize the denominator of the second expression: x^2 + 3x - 18.
This quadratic expression can be factored as (x + 6)(x - 3).
Now that we have factored both expressions, we can cancel out any common factors between the numerator of the first expression and the denominator of the second expression, as well as any common factors between the denominator of the first expression and the numerator of the second expression.
Canceling out the common factors, the expression becomes:
[(4x + 3)(x + 6) / (4x + 3)(4x - 3)] * [(4x^2 + 9x - 9) / (x + 6)(x - 3)]
Now, we can simplify the expression further.
Canceling out (4x + 3) terms from the numerator and denominator, the expression becomes:
[(x + 6) / (4x - 3)] * [(4x^2 + 9x - 9) / (x + 6)(x - 3)]
Next, we can cancel out (x + 6) terms from the numerator and denominator, resulting in:
1 / (4x - 3) * [(4x^2 + 9x - 9) / (x - 3)]
Finally, we simplify further by multiplying the numerators and denominators:
(4x^2 + 9x - 9) / (4x - 3)(x - 3)
Therefore, the simplified expression is (4x^2 + 9x - 9) / (4x - 3)(x - 3).
There are no restrictions on the variable in this expression.
To multiply the given expression, we can follow these steps:
Step 1: Simplify both numerator and denominator individually.
4x^2 + 27x + 18 can be factored as (2x + 3)(2x + 6)
16x^2 - 9 can be factored using the difference of squares as (4x - 3)(4x + 3)
4x^2 + 9x - 9 cannot be factored any further.
x^2 + 3x - 18 can be factored as (x - 3)(x + 6)
Step 2: Cancel any common factors between the numerator and denominator.
We can cancel out the (4x - 3) term.
After canceling out the common factor (4x - 3) in the numerator and denominator, the expression simplifies to:
((2x + 3)(2x + 6) * (4x + 9x - 9)) / ((4x + 3)(x - 3)(x + 6))
Step 3: Multiply the remaining terms in the numerator and denominator.
(2x + 3)(2x + 6) * (4x + 9x - 9) = (4x^2 + 12x + 6x + 18) * (4x + 9x - 9) = (4x^2 + 18x + 18) * (4x + 9x - 9) = 16x^3 + 36x^2 - 36x + 72x^2 + 162x - 162 - 36x - 81x + 81 = 16x^3 + 108x^2 + 45x - 81
(4x + 3)(x - 3)(x + 6) = (4x^2 - 12x + 3x - 9)(x + 6) = (4x^2 - 9x - 9)(x + 6) = 4x^3 + 24x^2 - 9x^2 - 54x - 9x - 54 = 4x^3 + 15x^2 - 63x - 54
Final Answer:
The simplified expression is (16x^3 + 108x^2 + 45x - 81) / (4x^3 + 15x^2 - 63x - 54)
Restrictions on the variable:
We should check if there are any values of x that would make the denominator equal to zero. In this case, we need to solve (4x + 3)(x - 3)(x + 6) = 0 to find any restrictions.
Setting each factor equal to zero, we have:
4x + 3 = 0 --> 4x = -3 --> x = -3/4
x - 3 = 0 --> x = 3
x + 6 = 0 --> x = -6
Therefore, the restrictions on the variable are x ≠ -3/4, x ≠ 3, and x ≠ -6.