Perform the following operation and express in simplest form.
start fraction, 3, x, divided by, 2, x, plus, 2, end fraction, dot, start fraction, x, squared, minus, 1, divided by, x, squared, plus, 6, x, minus, 7, end fraction
2x+2
3x
⋅
x
2
+6x−7
x
2
−1
To solve this problem, we need to multiply the two fractions.
First, we multiply the numerators together:
3x * (x^2 - 1) = 3x^3 - 3x
Next, we multiply the denominators together:
2x * (x^2 + 6x - 7) = 2x^3 + 12x^2 - 14x
So, our fraction becomes:
(3x^3 - 3x) / (2x^3 + 12x^2 - 14x)
We can't simplify this fraction any further because none of the terms in the numerator can be factored out of the terms in the denominator.
To simplify the given expression, we can follow these steps:
Step 1: Multiply the numerator of the first fraction by the numerator of the second fraction.
Result: 3x * (x^2 - 1)
Step 2: Multiply the denominator of the first fraction by the denominator of the second fraction.
Result: (2x + 2) * (x^2 + 6x - 7)
Step 3: Simplify the expression by canceling common factors, if any exist.
Thus, the expression, when simplified, is:
(start fraction) 3x * (x^2 - 1) divided by (2x + 2) * (x^2 + 6x - 7) (end fraction)
To simplify the expression, we need to perform the operation indicated (division) and express the result in its simplest form.
1. Start by multiplying the numerator (3x) by the reciprocal of the denominator (x^2 + 6x - 7). The reciprocal of a fraction is obtained by interchanging the numerator and the denominator.
So the reciprocal of (x^2 + 6x - 7) is (1 / (x^2 + 6x - 7)).
2. Now we can rewrite the expression as a multiplication:
(3x) * (1 / (x^2 + 6x - 7))
3. To multiply fractions, multiply the numerators and then multiply the denominators. The resulting fraction will be the new numerator, and the resulting denominator will be the new denominator.
So, (3x) * (1 / (x^2 + 6x - 7)) = (3x / 1) * (1 / (x^2 + 6x - 7))
4. Multiplying both numerators and denominators together, we get:
(3x * 1) / ((x^2 + 6x - 7) * 1)
5. Simplify further by canceling out common factors. In this case, there are no common factors, so we leave the expression as it is:
(3x) / (x^2 + 6x - 7)
Therefore, after performing the division and expressing the result in its simplest form, the given expression is:
(3x) / (x^2 + 6x - 7)