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
To find the quotient in simplest form, we need to simplify the expression. Let's break down the steps:
Step 1: Factor the given expressions
The numerator can be factored as the difference of squares:
x^2 - 4 = (x - 2)(x + 2)
The denominator in the first fraction can be factored as the difference of squares:
x - 3 = (x - 3)(1)
The numerator can be factored as the difference of squares:
x + 2 = (x + 2)(1)
The denominator in the second fraction can be factored as a quadratic trinomial:
x^2 + x - 12 = (x + 4)(x - 3)
Step 2: Simplify the expression
Now that we have factored the expressions, we can cancel out common factors.
The expression becomes:
((x - 2)(x + 2))/((x - 3)(x + 2)(x + 4))
Step 3: State any restrictions on the variable
Since we have denominators (x - 3) and (x + 2), we need to ensure that the values of x do not make these denominators equal to zero.
Therefore, the restrictions on the variable are:
x ≠ 3 (to avoid division by zero in the denominator (x - 3))
x ≠ -2 (to avoid division by zero in the denominator (x + 2))
Overall, the quotient in simplest form is ((x - 2)(x + 2))/((x - 3)(x + 2)(x + 4)) with the restrictions x ≠ 3 and x ≠ -2.
To simplify the quotient (x^2-4)/(x-3) divided by (x+2)/(x^2+x-12), we will first factorize the expressions:
x^2 - 4 = (x+2)(x-2)
x^2 + x - 12 = (x+4)(x-3)
Now, we can rewrite the expression as:
((x+2)(x-2))/(x-3) ÷ (x+2)/((x+4)(x-3))
Next, we can simplify the expression by multiplying the numerator and denominator of the first fraction by the reciprocal of the second fraction:
((x+2)(x-2))/(x-3) * ((x+4)(x-3))/(x+2)
Simplifying further, we can cancel out the common factors (x-3) and (x+2):
(x-2)(x+4)/(x+2)
So, the simplified expression is: (x-2)(x+4)/(x+2)
Now, let's state the restrictions on the variable. Since the denominator cannot be equal to zero, we have two restrictions:
1) x ≠ 3
2) x ≠ -2
Therefore, the restrictions on the variable are x ≠ 3 and x ≠ -2.
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.