Simplify the expression completely if possible.
start fraction, 3, x, cubed, minus, 6, x, squared, divided by, x, squared, plus, x, minus, 6, end fraction
x
2
+x−6
3x
3
−6x
2
To simplify the expression completely, we can first factor the numerator and the denominator:
Numerator:
3x^3 - 6x^2
Factoring out an x^2, we get:
x^2(3x - 6)
Simplifying the expression inside the parentheses, we have:
3x - 6 = 3(x - 2)
Therefore, the numerator can be written as:
x^2 * 3(x - 2) = 3x^2(x - 2)
Denominator:
x^2 + x - 6
Factoring the quadratic, we have:
(x + 3)(x - 2)
Now we can rewrite the expression:
(3x^2(x - 2))/((x + 3)(x - 2))
We can cancel out the common factors in the numerator and denominator:
(3x^2)/((x + 3))
Therefore, the simplified expression is:
(3x^2)/(x + 3)
To simplify the expression completely, we can start by factoring the numerator and denominator to see if there are any common factors we can cancel out.
The numerator, 3x^3 - 6x^2, can be factored by taking out the greatest common factor, which is 3x^2. Then, the numerator can be rewritten as 3x^2(x - 2).
The denominator, x^2 + x - 6, can be factored into (x + 3)(x - 2).
Now, we can cancel out the common factor of (x - 2) in both the numerator and denominator.
The simplified expression becomes:
(start fraction 3x^2)/(x + 3)
Note that we cannot further simplify this expression as there are no common factors left to cancel.
To simplify the expression, we can start by factoring the numerator and the denominator separately.
1. Factoring the numerator:
The numerator of the expression is 3x^3 - 6x^2. We can factor out the greatest common factor, which is 3x^2:
3x^3 - 6x^2 = 3x^2(x - 2)
2. Factoring the denominator:
The denominator of the expression is x^2 + x - 6. We can factor it by looking for two numbers that multiply to give -6 and add up to give 1 (the coefficient of x). The numbers 3 and -2 satisfy these conditions:
x^2 + x - 6 = (x + 3)(x - 2)
Now, the expression becomes:
(3x^2(x - 2)) / ((x + 3)(x - 2))
Notice that (x - 2) appears in both the numerator and denominator. We can cancel it out:
(3x^2) / (x + 3)
Therefore, the simplified expression is:
3x^2 / (x + 3)