Multiply. Show 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 expressions:
(4x^2 + 21x + 5) / (16x^2 - 1) * (4x^2 + 3x - 1) / (x^2 + 4x - 5)
First, factorize the denominators:
(4x^2 + 21x + 5) = (4x + 1)(x + 5)
(16x^2 - 1) = (4x + 1)(4x - 1)
(4x^2 + 3x - 1) = (4x - 1)(x + 1)
(x^2 + 4x - 5) = (x + 5)(x - 1)
Now, you can cancel out common factors:
(4x + 1) in the denominator of the first expression cancels out with (4x + 1) in the numerator of the second expression.
(x + 5) in the denominator of the first expression cancels out with (x + 5) in the numerator of the second expression.
(4x - 1) in the numerator of the first expression cancels out with (4x - 1) in the denominator of the second expression.
After canceling the common factors, we are left with:
(x + 1) / (4x - 1) * 4x^2 / (x - 1)
Multiplying the numerators and denominators separately:
(x + 1) * 4x^2 = 4x^3 + 4x^2
(4x - 1) * (x - 1) = 4x^2 - 4x - x + 1 = 4x^2 - 5x + 1
The final expression is:
(4x^3 + 4x^2) / (4x^2 - 5x + 1)
The restrictions of the answer are any values of x that would make the denominator equal to zero. So, for this expression, the restrictions are where:
4x^2 - 5x + 1 = 0
To find the restrictions, solve the quadratic equation:
4x^2 - 5x + 1 = 0
Using the quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a), where a = 4, b = -5, c = 1:
x = (-(-5) ± √((-5)^2 - 4*4*1))/(2*4)
x = (5 ± √(25 - 16))/8
x = (5 ± √9)/8
x = (5 ± 3)/8
So the restrictions for x are:
x = (5 + 3)/8 = 8/8 = 1
x = (5 - 3)/8 = 2/8 = 1/4
Therefore, the restrictions for the answer are x ≠ 1 and x ≠ 1/4.
To multiply the given expressions, let's follow these steps:
Step 1: Factorize the numerator and denominator of the first expression.
The numerator (4x^2 + 21x + 5) can be factored as (4x + 1)(x + 5).
The denominator (16x^2 - 1) can be factored as (4x + 1)(4x - 1).
So, the first expression becomes (4x + 1)(x + 5) / (4x + 1)(4x - 1).
Step 2: Factorize the numerator and denominator of the second expression.
The numerator (4x^2 + 3x - 1) cannot be factorized further.
The denominator (x^2 + 4x - 5) can be factored as (x + 5)(x - 1).
So, the second expression becomes (4x^2 + 3x - 1) / (x + 5)(x - 1).
Step 3: Cancel out common factors.
We can cancel out (4x + 1) from the numerator and denominator of the first expression. Also, (x + 5) can be canceled out from the numerator and denominator of both expressions. Therefore, the expression after canceling out common factors becomes:
(4x + 1)(x + 5) / (4x - 1)(x - 1).
Step 4: Multiply the numerators and denominators.
To multiply the numerators, we multiply the factors together:
(4x + 1)(x + 5) * (4x^2 + 3x - 1).
This simplifies to: 4x^3 + 27x^2 + 24x - 5.
To multiply the denominators, we multiply the factors together:
(4x - 1)(x - 1).
This simplifies to: 4x^2 - x - 4.
Step 5: Write down the final answer.
Putting it all together, the answer is:
(4x^3 + 27x^2 + 24x - 5) / (4x^2 - x - 4).
The restrictions on the answer are that the denominator (4x^2 - x - 4) cannot be equal to zero, as this would result in division by zero.
To multiply the given expression: (4x^2 + 21x + 5) / (16x^2 - 1) * (4x^2 + 3x - 1) / (x^2 + 4x - 5), we need to follow these steps:
Step 1: Factorize any quadratic expressions.
Step 2: Cancel out any common factors in the numerator and denominator.
Step 3: Multiply the remaining factors in both the numerator and denominator.
Let's go through each step.
Step 1: Factorize the quadratic expressions.
The first quadratic expression, 16x^2 - 1, is a difference of squares and can be factored as (4x - 1)(4x + 1).
The second quadratic expression, x^2 + 4x - 5, can be factored as (x - 1)(x + 5).
Now our expression becomes: (4x^2 + 21x + 5) / (4x - 1)(4x + 1) * (4x^2 + 3x - 1) / (x - 1)(x + 5).
Step 2: Cancel out common factors.
We notice that (4x - 1) and (x - 1) are common factors in the numerator and denominator. We can cancel them out.
The expression simplifies to: (4x^2 + 21x + 5) / (4x + 1) * (4x^2 + 3x - 1) / (x + 5).
Step 3: Multiply the remaining factors.
Multiply the numerators: (4x^2 + 21x + 5) * (4x^2 + 3x - 1) = 16x^4 + 12x^3 - 4x^2 + 84x^3 + 63x^2 - 21x + 20x^2 + 15x - 5.
Multiply the denominators: (4x + 1) * (x + 5) = 4x^2 + 20x + x + 5 = 4x^2 + 21x + 5.
Now, the expression becomes: (16x^4 + 12x^3 - 4x^2 + 84x^3 + 63x^2 - 21x + 20x^2 + 15x - 5) / (4x^2 + 21x + 5).
The restrictions of the answer occur when the denominator equals zero since division by zero is undefined. To find these restrictions, set the denominator equal to zero and solve the equation:
4x^2 + 21x + 5 = 0.
You can solve this quadratic equation by factoring or using the quadratic formula. Once you find the solutions for x, those are the restrictions for this expression.