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.
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
1 answer