Questions: Simplify the rational expression. 2x+2 / 2x^3+2 A. 1 / x^2-x+1 B. 1 / x^2+x+1 C. 1 / x^2+1 D. 1 / x^2

Solution Steps

To simplify the given rational expression, we first factor out the common terms in the numerator and the denominator. The numerator \(2x + 2\) can be factored as \(2(x + 1)\). The denominator \(2x^3 + 2\) can be factored as \(2(x^3 + 1)\). We then simplify the expression by canceling the common factor of 2.

We start with the rational expression:

\[ \frac{2x + 2}{2x^3 + 2} \]

We can factor the numerator:

\[ 2x + 2 = 2(x + 1) \]

And the denominator:

\[ 2x^3 + 2 = 2(x^3 + 1) \]

Now we can rewrite the expression using the factored forms:

\[ \frac{2(x + 1)}{2(x^3 + 1)} \]

We can cancel the common factor of 2 from the numerator and the denominator:

\[ \frac{x + 1}{x^3 + 1} \]

The denominator \(x^3 + 1\) can be factored using the sum of cubes formula:

\[ x^3 + 1 = (x + 1)(x^2 - x + 1) \]

Thus, we can rewrite the expression as:

\[ \frac{x + 1}{(x + 1)(x^2 - x + 1)} \]

Now, we can cancel the common factor \(x + 1\) (assuming \(x \neq -1\)):

\[ \frac{1}{x^2 - x + 1} \]

Final Answer