Multiply. State all restictions. Show your work.
4x^2+27x+18/16x^2-9 * 4x^2+9x-9/x^2+3x-18
To multiply the given expressions, we need to simplify each expression and then multiply the simplified forms.
First, let's simplify the expression (4x^2 + 27x + 18) / (16x^2 - 9):
We notice that both the numerator and denominator are perfect squares, since they can be factored as the difference of squares.
(4x^2 + 27x + 18) / (16x^2 - 9)
= [(2x + 3)(2x + 6)] / [(4x + 3)(4x - 3)]
= (2x + 3)(2x + 6) / (4x + 3)(4x - 3)
Next, let's simplify the expression (4x^2 + 9x - 9) / (x^2 + 3x - 18):
We factor the numerator and denominator to see if there is any common factor to cancel.
(4x^2 + 9x - 9) / (x^2 + 3x - 18)
= (2x - 3)(2x + 3) / (x - 3)(x + 6)
Now, we can multiply the simplified forms:
[(2x + 3)(2x + 6) / (4x + 3)(4x - 3)] * [(2x - 3)(2x + 3) / (x - 3)(x + 6)]
When multiplying fractions, we can cancel out common factors appearing on the numerator of one fraction and the denominator of the other fraction. In this case, we have (2x + 3) and (2x - 3) as common factors.
[(2x + 6) / (4x + 3)(4x - 3)] * [(2x + 3) / (x - 3)(x + 6)]
The final result is:
(2x + 6)(2x + 3) / (4x + 3)(4x - 3)(x - 3)(x + 6)
Restrictions:
We need to consider any values of x which would make the denominator equal to zero since division by zero is undefined.
The restrictions are:
4x + 3 = 0, which gives x = -3/4
4x - 3 = 0, which gives x = 3/4
x - 3 = 0, which gives x = 3
x + 6 = 0, which gives x = -6
Therefore, the restrictions are x ≠ -3/4, x ≠ 3/4, x ≠ 3, and x ≠ -6.
To multiply the given expression, first factor both the numerator and denominator.
The numerator 4x^2 + 27x + 18 factors into (2x + 3)(2x + 6).
The denominator 16x^2 - 9 factors into (4x + 3)(4x - 3).
The numerator 4x^2 + 9x - 9 does not factor further.
The denominator x^2 + 3x - 18 can be factored as (x + 6)(x - 3).
Now, we can cancel out the common factors if any.
Canceling out the (2x + 3) factor from the numerator and the (4x + 3) factor from the denominator leaves us with:
(2x + 6)/(4x - 3) * (4x^2 + 9x - 9)/((x + 6)(x - 3))
Now, multiply the numerators together and the denominators together:
(2x + 6) * (4x^2 + 9x - 9) / (4x - 3) * (x + 6)(x - 3)
Expanding the numerator, we get:
(2x + 6) * (4x^2 + 9x - 9) = 8x^3 + 6x^2 + 18x^2 + 54x - 18x - 54
= 8x^3 + 24x^2 +36x - 54
The denominator remains the same:
(4x - 3) * (x + 6)(x - 3) = (4x - 3)(x^2 + 3x - 18)
= 4x(x^2 + 3x - 18) - 3(x^2 + 3x - 18)
= 4x^3 + 12x^2 - 72x - 3x^2 - 9x + 54
= 4x^3 + 9x^2 - 81x + 54
Combining both the numerator and denominator, we get:
(8x^3 + 24x^2 + 36x - 54) / (4x^3 + 9x^2 - 81x + 54)
The final expression after multiplication is:
(8x^3 + 24x^2 + 36x - 54) / (4x^3 + 9x^2 - 81x + 54)
There are no specific restrictions mentioned in the problem.
To multiply these two rational expressions, we'll first factor both the numerators and denominators. Then, we'll cancel out any common factors between the numerators and denominators to simplify the expression. Finally, we'll multiply the remaining factors in the numerator and denominator to obtain the simplified expression.
Given expression:
(4x^2 + 27x + 18) / (16x^2 - 9) * (4x^2 + 9x - 9) / (x^2 + 3x - 18)
Step 1: Factor the numerators and denominators.
For the first rational expression:
Numerator: 4x^2 + 27x + 18
Factors: (2x + 3)(2x + 6)
Denominator: 16x^2 - 9
Factors: (4x + 3)(4x - 3)
For the second rational expression:
Numerator: 4x^2 + 9x - 9
This expression is already in factored form.
Denominator: x^2 + 3x - 18
Factors: (x + 6)(x - 3)
Step 2: Cancel out common factors between the numerators and denominators.
We can cancel out (2x + 3) and (4x - 3) between the first rational expression's numerator and denominator.
Step 3: Multiply the remaining factors.
Numerator: (2x + 6) * (4x^2 + 9x - 9)
= 8x^3 + 18x^2 - 18x + 24x^2 + 54x - 54
= 8x^3 + 42x^2 + 36x - 54
Denominator: (4x + 3) * (x + 6) * (x - 3)
= (4x + 3) * (x^2 + 3x - 18)
= 4x^3 + 12x^2 - 72x + 3x^2 + 9x - 54
= 4x^3 + 15x^2 - 63x - 54
Step 4: Write the final simplified expression.
The simplified expression after multiplying the two rational expressions is:
(8x^3 + 42x^2 + 36x - 54) / (4x^3 + 15x^2 - 63x - 54)
Restrictions:
To determine any restrictions, we need to find the values of x that make the denominator equal to zero. This is because dividing by zero is undefined and not allowed in mathematics. So, we set the denominator equal to zero and solve for x.
Denominator: 4x^3 + 15x^2 - 63x - 54
Setting this equal to zero, we get:
4x^3 + 15x^2 - 63x - 54 = 0
By factoring or using algebraic methods, we can determine the possible values of x that make the denominator zero. In this case, the restrictions are the values of x which make the equation equal to zero.