Multiply. State any restrictions on the variable.
x^2-5x+6/x^2-4 * x^2+3x+2/x^2-2x-3
To multiply the two rational expressions, we can first factor the numerators and denominators:
(x^2-5x+6)/(x^2-4) * (x^2+3x+2)/(x^2-2x-3)
(x-3)(x-2)/(x-2)(x+2) * (x+2)(x+1)/(x+1)(x-3)
Now we can cancel out the common factors in the numerator and denominator:
(x-3)/(x+2) * 1/(x+1)
The final expression is (x-3)/(x+2) * 1/(x+1).
There are no restrictions on the variable, as long as the denominators are non-zero.
To multiply the given expressions, we will first factorize the numerators and denominators, and then cancel out common factors:
Numerator 1: x^2 - 5x + 6
Factoring numerator 1: (x - 2)(x - 3)
Denominator 1: x^2 - 4
Factoring denominator 1: (x + 2)(x - 2)
Numerator 2: x^2 + 3x + 2
Factoring numerator 2: (x + 2)(x + 1)
Denominator 2: x^2 - 2x - 3
Factoring denominator 2: (x - 3)(x + 1)
Now, we can multiply the numerators together and the denominators together:
Numerator: (x - 2)(x - 3) * (x + 2)(x + 1) = (x^2 - 5x + 6)(x^2 + 3x + 2)
Denominator: (x + 2)(x - 2) * (x - 3)(x + 1) = (x^2 - 4)(x^2 - 2x - 3)
Finally, we can write the multiplication result as:
(x^2 - 5x + 6)(x^2 + 3x + 2) / (x^2 - 4)(x^2 - 2x - 3)
Restrictions on the variables:
From the denominators, we observe that both expressions will be undefined if any of the denominators equal zero. Thus, the restrictions on the variable are:
x^2 - 4 = 0 => x = -2 or x = 2
x^2 - 2x - 3 = 0 => x = -1 or x = 3
Hence, x cannot be equal to -2, 2, -1, or 3.
To multiply the two fractions, we can follow these steps:
Step 1: Factorize the numerators and denominators of both fractions:
Numerator of the first fraction: x^2 - 5x + 6 = (x - 2)(x - 3)
Denominator of the first fraction: x^2 - 4 = (x - 2)(x + 2)
Numerator of the second fraction: x^2 + 3x + 2 = (x + 1)(x + 2)
Denominator of the second fraction: x^2 - 2x - 3 = (x - 3)(x + 1)
Step 2: Cancel out any common factors in the numerators and denominators.
In this case, we can cancel out (x - 2), (x - 3), and (x + 1) from both the numerator and denominator.
After canceling out the common factors, we are left with:
[(x - 3)(x + 1)] / [(x + 2)(x - 3)]
Step 3: Simplify the expression if possible.
Since we canceled out the common factor (x - 3) from the numerator and denominator, we can simplify the expression to:
(x + 1) / (x + 2)
Restrictions on the variable:
From the original expression, we can see that denominators are x^2 - 4 and x^2 - 2x - 3. To find any restrictions on the variable, we need to look for values of x that would make these denominators equal to zero.
For the first denominator, x^2 - 4 = 0, we solve for x as follows:
x^2 = 4
x = ±2
So the variable x cannot be equal to 2 or -2 to avoid division by zero for the first fraction.
For the second denominator, x^2 - 2x - 3 = 0, we solve for x using the quadratic formula:
x = (-(-2) ± √((-2)^2 - 4(1)(-3))) / (2(1))
x = (2 ± √(4 + 12)) / 2
x = (2 ± √16) / 2
x = (2 ± 4) / 2
x = (2 + 4) / 2 or x = (2 - 4) / 2
x = 6 / 2 or x = -2 / 2
x = 3 or x = -1
So the variable x cannot be equal to 3 or -1 to avoid division by zero for the second fraction.
Therefore, the restrictions on the variable are x ≠ 2, x ≠ -2, x ≠ 3, and x ≠ -1.