How do you multiply #\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18}#?

Question

Ethan Adkins · Accepted Answer

#color(indigo)(=> ((3x+2)(x+6)) / ((x-1)(x-6)) = (3x^2 + 20x + 12) / (x^2 - 7x + 6)# #12x^2 - x - 6 = 12x^2 - 9x + 8x - 6 = 3x(4x - 3) + 2 (4x - 3)# #=> (3x+2) (4x-3)# #x^2 + 7x = 6 = x^2 + 6x + x + 6 = (x + 6) (x+1)# #4x^2 - 27x + 18 = 4x^2 - 24x - 3x + 18 = (x-6) (4x-3)# #((12x^2 - x - 6)/(x^2-1)) * ((x^2 + 7x + 6) / (4x^2 - 27x + 18))# #=> ((3x+2)cancel((4x-3)))/(cancel((x+1))(x-1)) * ((x+6)cancel((x+1)))/((x-6)cancel((4x-3)))# #color(indigo)(=> ((3x+2)(x+6)) / ((x-1)(x-6)) = (3x^2 + 20x + 12) / (x^2 - 7x + 6)#

Kennedy Adams · Answer

undefined To multiply the given expressions, we can follow these steps:

1. Factorize both numerator and denominator of each fraction, if possible.
2. Multiply the numerators together to get the new numerator.
3. Multiply the denominators together to get the new denominator.
4. Simplify the resulting fraction, if possible.

Let's apply these steps to the given expression:

1. Factorize the first fraction:
   Numerator: 12x^2 - x - 6 = (3x - 2)(4x + 3)
   Denominator: x^2 - 1 = (x - 1)(x + 1)

Factorize the second fraction:
   Numerator: x^2 + 7x + 6 = (x + 1)(x + 6)
   Denominator: 4x^2 - 27x + 18 = (4x - 3)(x - 6)

2. Multiply the numerators:
   (3x - 2)(4x + 3) * (x + 1)(x + 6)

3. Multiply the denominators:
   (x - 1)(x + 1) * (4x - 3)(x - 6)

4. Simplify the resulting fraction, if possible.

Therefore, the multiplication of the given expressions is:
(3x - 2)(4x + 3)(x + 1)(x + 6) / (x - 1)(x + 1)(4x - 3)(x - 6)