What is the simplified form of the quantity 12 z-squared minus 25z plus 12 over the quantity 3 z-squared plus 2z minus 8

If possible, please convert these words to the equivalent symbolic equation:

12z^2 - 25z + 12

3z^2 + 2z - 8

It's possible that the numerator and the denominator share a common factor. Thus, the first thing we should do here is try to factor the simpler 3z^2 + 2z - 8:

(3z - 4)(z + 2 ).

Next, determine whether either of these two factors is also a factor of

12z^2 - 25z + 12. Borrowing the factor (3z-4) and noting that 4/3 would be the corresponding divisor in synthetic division, I found that there was a zero remainder, which tells us that (3z - 4) actually is a factor of 12z^2 - 25z + 12.

Thus, the original expression,

12z^2 - 25z + 12 (3z-4)(4z-3)

factors into

3z^2 + 2z - 8 (3z-4)(z+2)

The common factor here is (3z-4). Cancelling that, we get:

4z-3

which is the final answer, the quotient reduced to lowest terms.

z+2

                                     cos(4u) = cos²(2u) - sin²(2u)
                             cos(2u + 2u) = cos²(2u) - sin²(2u)
cos(2u)cos(2u) - sin(2u)sin(2u) = cos²(2u) - sin²(2u)
                     cos²(2u) - sin²(2u) = cos²(2u) - sin²(2u)