Multiply. State all restictions. Show your work.
4x^2+21x+5/16x^2-1 * 4x^2+3x-1/x^2+4x-5
To multiply the expression (4x^2+21x+5)/(16x^2-1) * (4x^2+3x-1)/(x^2+4x-5), we can simplify each numerator and denominator separately before multiplying.
First, let's simplify the first fraction (4x^2+21x+5)/(16x^2-1).
The numerator (4x^2+21x+5) cannot be simplified further.
For the denominator (16x^2-1), it is a difference of squares, so we can factor it to:
(4x+1)(4x-1).
Now let's simplify the second fraction (4x^2+3x-1)/(x^2+4x-5).
The numerator (4x^2+3x-1) cannot be simplified further.
The denominator (x^2+4x-5) can be factored to:
(x+5)(x-1).
Now we have (4x^2+21x+5)/(16x^2-1) * (4x^2+3x-1)/(x^2+4x-5) = ((4x^2+21x+5)/(4x+1)(4x-1)) * ((4x^2+3x-1)/(x+5)(x-1)).
Next, we can cancel out common factors between the numerators and denominators. We have the following cancellations:
(4x^2 + 21x + 5)/(4x+1) * (4x^2+3x-1)/(x+5)
Now we can multiply the numerators and denominators together:
(4x^2 + 21x + 5)(4x^2 + 3x - 1) / (4x + 1)(x + 5)
Expanding the numerator:
(16x^4 + 12x^3 - 4x^2 + 84x^3 + 63x^2 - 21x + 20x^2 + 15x - 5) / (4x + 1)(x + 5)
Combining like terms in the numerator:
(16x^4 + 96x^3 + 79x^2 + 57x - 5) / (4x + 1)(x + 5)
Therefore, the simplified expression is (16x^4 + 96x^3 + 79x^2 + 57x - 5) / (4x + 1)(x + 5).
There are no additional restrictions for this multiplication since there are no common factors that could lead to division by zero.
To multiply the given expressions, we first need to factor each expression to simplify the multiplication. Let's start by factoring each expression:
Expression 1: 4x^2 + 21x + 5
Factors as: (2x + 1)(2x + 5)
Expression 2: 16x^2 - 1
Factors as: (4x + 1)(4x - 1)
Expression 3: 4x^2 + 3x - 1
Doesn't factor further.
Expression 4: x^2 + 4x - 5
Factors as: (x + 5)(x - 1)
Now, we can rewrite the multiplication using the factored forms:
(4x^2 + 21x + 5) / (16x^2 - 1) * (4x^2 + 3x - 1) / (x^2 + 4x - 5)
= [(2x + 1)(2x + 5)] / [(4x + 1)(4x - 1)] * (4x^2 + 3x - 1) / [(x + 5)(x - 1)]
Next, we can cancel out any common factors between the numerators and denominators:
= [(2x + 1)(2x + 5)(4x^2 + 3x - 1)] / [(4x + 1)(4x - 1)(x + 5)(x - 1)]
The restrictions for this expression can be determined by identifying any values of x that would result in a zero denominator. In this case, the denominators (4x + 1), (4x - 1), (x + 5), and (x - 1) should not be equal to zero. So we have the following restrictions:
4x + 1 ≠ 0
4x - 1 ≠ 0
x + 5 ≠ 0
x - 1 ≠ 0
Solving each of these inequalities, we get:
4x ≠ -1
4x ≠ 1
x ≠ -5
x ≠ 1
Therefore, the restrictions are x ≠ -1/4, x ≠ 1/4, x ≠ -5, and x ≠ 1.
That's the step-by-step process of multiplying the given expressions and finding the restrictions.
To multiply the given expressions, we can follow these steps:
Step 1: Factorize the numerators and denominators of both fractions to simplify the expressions if possible.
Numerator 1: 4x^2 + 21x + 5
This expression cannot be factored further.
Numerator 2: 4x^2 + 3x - 1
This expression cannot be factored further.
Denominator 1: 16x^2 - 1
This expression is a difference of squares and can be factored as (4x + 1)(4x - 1).
Denominator 2: x^2 + 4x - 5
This expression can be factored as (x + 5)(x - 1).
After simplifying, we have the following expression:
(4x^2 + 21x + 5) / (16x^2 - 1) * (4x^2 + 3x - 1) / (x^2 + 4x - 5)
Step 2: Cancel out common factors between the numerators and denominators.
(4x^2 + 21x + 5) / (16x^2 - 1) * (4x^2 + 3x - 1) / (x^2 + 4x - 5)
can be rewritten as
[(4x^2 + 21x + 5) * (4x^2 + 3x - 1)] / [(16x^2 - 1) * (x^2 + 4x - 5)]
Step 3: Multiply the numerators and denominators separately.
(4x^2 + 21x + 5) * (4x^2 + 3x - 1)
= 16x^4 + 12x^3 - 4x^2 + 84x^3 + 63x^2 - 21x + 20x^2 + 15x - 5
= 16x^4 + 96x^3 + 79x^2 + 57x - 5
(16x^2 - 1) * (x^2 + 4x - 5)
= 16x^4 + 64x^3 - 80x^2 - x^2 - 4x + 5
= 16x^4 + 64x^3 - 81x^2 - 4x + 5
Step 4: Combine the multiplied expressions and simplify, if possible.
[(4x^2 + 21x + 5) * (4x^2 + 3x - 1)] / [(16x^2 - 1) * (x^2 + 4x - 5)]
= (16x^4 + 96x^3 + 79x^2 + 57x - 5) / (16x^4 + 64x^3 - 81x^2 - 4x + 5)
There are no restrictions in this multiplication problem.