Question

The quotient of (x^3+3x^2-4x-12)/(x^2+5x+6)

Answer 1

x-2 ....................

Answer 2

Answer: The quotient is (x-2).

Explanation:

Since we have given that

`f(x)=(x^3+3x^2-4x-12)\\\\and\\\\g(x)=x^2+5x+6\\\\So,\ (\left(x^3+3x^2-4x-12\right))/(\left(x^2+5x+6\right))`

Now, we have to find the quotient of the above expression.

So, here we go:

`Factorise\ (x^3+3x^2-4x-12)\\\\=\left(x^3+3x^2\right)+\left(-4x-12\right)\\\\=-4\left(x+3\right)+x^2\left(x+3\right)\\\\=\left(x+3\right)\left(x^2-4\right)`

Now, we will divide the above simplest form with g(x):

`(\left(x+3\right)\left(x^2-4\right))/(\left(x+2\right)\left(x+3\right))\\\\=(x^2-4)/(x+2)\\\\=(\left(x+2\right)\left(x-2\right))/(x+2)\ using\ (a^2-b^2)=(a+b)(a-b)\\\\=x-2`

Hence, the quotient is (x-2).