Multiply. State all restictions of the answer. Show your work.
4x^2+21x+5/16x^2-1 * 4x^2+3x-1/x^2+4x-5
To multiply the given expression, we will first factor the numerators and denominators separately:
Numerator 1: 4x^2 + 21x + 5 = (4x + 1)(x + 5)
Denominator 1: 16x^2 - 1 = (4x - 1)(4x + 1)
Numerator 2: 4x^2 + 3x - 1 = (4x - 1)(x + 1)
Denominator 2: x^2 + 4x - 5 = (x + 5)(x - 1)
Now, we can cancel out common factors and simplify the expression:
(4x + 1)(x + 5) / (4x - 1)(4x + 1) * (4x - 1)(x + 1) / (x + 5)(x - 1)
After canceling out common factors, we are left with:
(x + 5) / (4x - 1) * (x + 1) / (x - 1)
Now, we can multiply the fractions by multiplying the numerators and denominators:
(x + 5)(x + 1) / (4x - 1)(x - 1)
To state the restrictions of the answer, we need to consider any values that would make the denominators equal to zero. In this case, we have:
(4x - 1)(x - 1) = 0
From this, we can find the restrictions:
4x - 1 = 0 --> 4x = 1 --> x = 1/4
x - 1 = 0 --> x = 1
Therefore, the restrictions of the answer are x = 1/4 and x = 1.
To multiply the given expressions, we'll first factorize the numerator and denominator of each fraction. Then we'll cancel out any common factors. Finally, we'll multiply the remaining factors together. Let's go step by step.
Step 1: Factorize the expressions:
Numerator:
4x^2 + 21x + 5 is already prime, so we can't factorize it any further.
Denominator:
16x^2 - 1 can be written as (4x + 1)(4x - 1).
4x^2 + 3x - 1 is already prime, so we can't factorize it any further.
x^2 + 4x - 5 can be written as (x + 5)(x - 1).
Step 2: Cancel out common factors:
The (4x - 1) term appears in both the numerator and denominator, so we can cancel it out.
Step 3: Multiply the remaining factors together:
After canceling out the common factors, we are left with the following expression:
(4x^2 + 21x + 5)/(16x^2 - 1) * (4x^2 + 3x - 1)/(x^2 + 4x - 5)
= [(4x^2 + 21x + 5)/(16x^2 - 1)] * [(4x^2 + 3x - 1)/(x^2 + 4x - 5)]
= [(4x + 1)(x + 5)/(4x + 1)(4x - 1)] * [(4x^2 + 3x - 1)/(x + 5)(x - 1)]
= [(x + 5)/(4x - 1)] * [(4x^2 + 3x - 1)/(x + 5)(x - 1)]
= [(4x^2 + 3x - 1)/(4x - 1)(x - 1)]
Answer: The simplified expression after multiplying is (4x^2 + 3x - 1)/(4x - 1)(x - 1). The restrictions would be that (4x - 1) and (x - 1) should not equal zero to avoid division by zero. So, x cannot be 1 or 1/4.
To multiply the given expressions, you need to multiply the numerators together and multiply the denominators together separately. Let's break it down step by step.
First, we'll multiply the numerators: (4x^2 + 21x + 5) * (4x^2 + 3x - 1).
To do this, you can use the distributive property or apply the FOIL method.
(4x^2 + 21x + 5) * (4x^2 + 3x - 1)
= 4x^2 * 4x^2 + 4x^2 * 3x + 4x^2 * (-1) + 21x * 4x^2 + 21x * 3x + 21x * (-1) + 5 * 4x^2 + 5 * 3x + 5 * (-1)
Multiplying each term and combining like terms yields:
= 16x^4 + 12x^3 - 4x^2 + 84x^3 + 63x^2 - 21x + 20x^2 + 15x - 5
= 16x^4 + (12x^3 + 84x^3) + (-4x^2 + 63x^2 + 20x^2) + (-21x + 15x) - 5
= 16x^4 + 96x^3 + 79x^2 - 6x - 5
Next, we'll multiply the denominators: (16x^2 - 1) * (x^2 + 4x - 5).
Again, we can use the distributive property or simplify it further.
(16x^2 - 1) * (x^2 + 4x - 5)
= 16x^2 * x^2 + 16x^2 * 4x - 16x^2 * 5 - 1 * x^2 - 1 * 4x - 1 * (-5)
Multiplying each term and combining like terms, we get:
= 16x^4 + 64x^3 - 80x^2 - x^2 - 4x + 5
= 16x^4 + 64x^3 - (80 + 1)x^2 - 4x + 5
= 16x^4 + 64x^3 - 81x^2 - 4x + 5
Now, let's put together the numerator and denominator:
(16x^4 + 96x^3 + 79x^2 - 6x - 5) / (16x^4 + 64x^3 - 81x^2 - 4x + 5)
Finally, let's analyze the restrictions on the answer.
In the denominator, we have the expression (16x^4 + 64x^3 - 81x^2 - 4x + 5). To find the restrictions, we need to check where the denominator will be equal to zero. If it is, then those values will be excluded from the domain.
Setting the denominator equal to zero:
16x^4 + 64x^3 - 81x^2 - 4x + 5 = 0
Unfortunately, finding the exact solutions for this equation may be complex, and the restrictions depend on the roots of the denominator. To determine the restrictions definitively, it would be necessary to solve the equation.
Essentially, the restrictions of the answer are the values of "x" that would make the denominator zero, which are generally considered as non-permissible values in the domain.