What is the quotient in simplest form? State any restrictions on the variable. x^2-4/x-3 divided by x+2/x^2+x-12

Answer 1

To simplify the quotient, we can factor both the numerator and denominator before dividing.

Numerator: x^2 - 4 = (x + 2)(x - 2)
Denominator: x - 3
Numerator of the divisor: x + 2
Denominator of the divisor: x^2 + x - 12 = (x + 4)(x - 3)

Now, we can rewrite the quotient as a multiplication of the reciprocals:

((x + 2)(x - 2))/(x - 3) * (x^2 + x - 12)/(x + 2)

The common factors of x + 2 can be canceled out:

(x - 2)/(x - 3) * (x^2 + x - 12)

Next, we can multiply the remaining terms:

= (x - 2)(x^2 + x - 12)/(x - 3)

= (x^3 + x^2 - 12x - 2x^2 - 2x + 24)/(x - 3)

= (x^3 - x^2 - 14x + 24)/(x - 3)

Therefore, the quotient in simplest form is (x^3 - x^2 - 14x + 24)/(x - 3). The restriction on the variable is x ≠ 3, as the denominator (x - 3) cannot be zero.