What is the quotient in simplified form? State any restrictions on the variable.
X^2-16/ x^2+5x+6 divided by x^2+5x+4/ x^2-2x-8
To divide by a fraction, we multiply by the reciprocal of the fraction. Therefore, we rewrite the division as a multiplication by flipping the second fraction:
(X^2 - 16) / (x^2 + 5x + 6) * (x^2 - 2x - 8) / (x^2 + 5x + 4)
Next, we factor the numerator and denominator of both fractions:
((x + 4)(x - 4)) / ((x + 3)(x + 2)) * ((x - 4)(x + 2)) / ((x + 4)(x + 1))
After factoring, we can simplify:
((x - 4)(x + 2)(x - 4)(x + 2)) / ((x + 3)(x + 2)(x + 4)(x + 1))
This simplifies to:
(x - 4)(x + 2) / (x + 3)(x + 4)
Therefore, the quotient, in simplified form, is:
(x - 4)(x + 2) / (x + 3)(x + 4)
The restrictions on the variable are that x cannot equal -3, -4, -1, or 4 to avoid division by zero.