Simplify:
x4 −4x3 −27x+108
----------------
x3 −3x2 −16x+48
PLEASE!
where do you get x3 from? do you mean 3x?
To simplify the given expression, let's start by factoring both the numerator and the denominator.
The numerator, x^4 - 4x^3 - 27x + 108, can be factored by grouping as follows:
x^4 - 4x^3 - 27x + 108 = (x^4 - 27x) + (-4x^3 + 108)
= x(x^3 - 27) - 4(x^3 - 27)
= x(x - 3)(x^2 + 3x + 9) - 4(x - 3)(x^2 + 3x + 9)
= (x - 3)(x(x^2 + 3x + 9) - 4(x^2 + 3x + 9))
= (x - 3)(x^3 + 3x^2 + 9x - 4x^2 - 12x - 36)
= (x - 3)(x^3 - x^2 - 3x - 36)
Now let's factor the denominator, x^3 - 3x^2 - 16x + 48:
x^3 - 3x^2 - 16x + 48 = (x^3 - 16x) + (-3x^2 + 48)
= x(x^2 - 16) - 3(x^2 - 16)
= x(x - 4)(x + 4) - 3(x - 4)(x + 4)
= (x - 4)(x(x + 4) - 3(x + 4))
= (x - 4)(x^2 + 4x - 3x - 12)
= (x - 4)(x^2 + x - 12)
Now that we have factored the numerator and the denominator, we can simplify the expression:
(x^4 - 4x^3 - 27x + 108)/(x^3 - 3x^2 - 16x + 48) = [(x - 3)(x^3 - x^2 - 3x - 36)] / [(x - 4)(x^2 + x - 12)]
Since we have the same factor, (x - 3), in both the numerator and the denominator, we can cancel them out:
(x^4 - 4x^3 - 27x + 108)/(x^3 - 3x^2 - 16x + 48) = (x^3 - x^2 - 3x - 36)/(x^2 + x - 12)
Therefore, the simplified expression is (x^3 - x^2 - 3x - 36)/(x^2 + x - 12).