(x^3+2x+4x)/(x^3+8)
To simplify the expression (x^3 + 2x + 4x)/(x^3 + 8), we can start by factoring the numerator and denominator.
Step 1: Factor the Numerator
In the numerator, there are two terms with x. We can simplify those terms by factoring out x:
x^3 + 2x = x(x^2 + 2)
Step 2: Factor the Denominator
In the denominator, x^3 + 8, we can recognize that this is a sum of cubes. Sum of cubes can be factored as follows:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
In our case, a is x and b is 2:
x^3 + 8 = (x + 2)(x^2 - 2x + 4)
Step 3: Simplify the Expression
Now that we have factored the numerator and denominator, we can simplify the expression:
(x^3 + 2x + 4x)/(x^3 + 8) = [x(x^2 + 2)] / [(x + 2)(x^2 - 2x + 4)]
And that is the simplified form of the expression (x^3 + 2x + 4x)/(x^3 + 8).