What is the quotient((6-x)/(x^2+2x-3))/((x^2-4x-12)/(x^2+4x+3)) in simplified form? State any restrictions on the variable.

Answer 1

To simplify the expression, first factor all the denominators and numerators:

(6-x)/(x^2 + 2x - 3) = (6 - x)/((x - 1)(x + 3))

(x^2 - 4x - 12) = (x - 6)(x + 2)

(x^2 + 4x + 3) = (x + 3)(x + 1)

Now rewrite the expression:

((6-x)/(x^2 + 2x - 3))/((x^2 - 4x - 12)/(x^2 + 4x + 3))

= [(6 - x) / ((x - 1)(x + 3))] / [((x - 6)(x + 2)) / ((x + 3)(x + 1))]

= [(6 - x) / ((x - 1)(x + 3))] * [((x + 3)(x + 1)) / ((x - 6)(x + 2))]

= [(6 - x)(x + 3)(x + 1)] / [(x - 1)(x + 3)(x - 6)(x + 2)]

Simplify the expression:

= [-x^3 + 6x^2 + 9x + 18] / [(- x^2 - 7x + 6)(x^2 - 4x - 12)]

Restrictions:
1. The denominators cannot be equal to 0. Therefore, x cannot be equal to 1, -3, 6, -2.
2. The expression is undefined for any value of x that makes the denominator of the initial expression become 0. These values are x = 1, -3.

Therefore, the simplified form of the quotient is (-x^3 + 6x^2 + 9x + 18) / (-x^2 - 7x + 6)(x^2 - 4x - 12) with restrictions x ≠ 1, -3, -2, 6.