What is the quotient in simplest form? state any restrictions on the variable
x^2/x+4 divided by x-2/x^2+x-12
I bet you meant:
x^2/(x+4) ÷ ( (x-2)/(x^2+x-12) )
= x^2/(x+4) * (x+4)(x-3)/(x-2)
= x^2(x-3)/(x-2) , x ≠ -4,2
To determine the quotient of the given expression in simplest form, we will divide the two fractions and simplify if possible.
First, let's rewrite the expression and put it in fraction form:
(x^2 / (x + 4)) / ((x - 2) / (x^2 + x - 12))
Now, recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the expression as follows:
(x^2 / (x + 4)) * ((x^2 + x - 12) / (x - 2))
Next, we need to factorize the quadratic expressions in the numerator and denominator:
(x^2 / (x + 4)) * ((x - 3)(x + 4) / (x - 2))
Now, we can cancel out the common factor of (x + 4) in the numerator and denominator:
(x^2 / 1) * ((x - 3) / (x - 2))
Finally, we multiply the remaining terms together:
x^2 * (x - 3) / (1 * (x - 2))
Simplifying the expression, we have:
(x^3 - 3x^2) / (x - 2)
Therefore, the quotient in simplest form is (x^3 - 3x^2) / (x - 2).
Restrictions on the variable:
To determine any restrictions on the variable, we need to identify values that would make the denominator equal to zero. In this case, if we set x - 2 = 0, solving for x gives us x = 2. Therefore, the expression is undefined when x = 2. Hence, the restriction on the variable is x ≠ 2.