((x ^ 2 + 2x + 1)/(x ^ 3 - 8x + 16))/((x + 1)/(x ^ 3 - 16))

1 answer

To simplify this complex fraction, we will need to follow these steps:

1. Rewrite the complex fraction as a division of two fractions.
2. Factor all the polynomials in the numerator and denominator.
3. Simplify the fraction by canceling common factors.

Let's apply these steps to the given expression:

1. Rewrite as a division of two fractions:

((x ^ 2 + 2x + 1)/(x ^ 3 - 8x + 16)) ÷ ((x + 1)/(x ^ 3 - 16))

2. Factor all the polynomials:

(x + 1) can be factored as (x + 1)

(x ^ 2 + 2x + 1) can be factored using the quadratic formula as (x + 1)^2

(x ^ 3 - 8x + 16) can be factored using long division or synthetic division as (x - 2)(x ^ 2 + 2x - 4)

(x ^ 3 - 16) can be factored using the sum of cubes formula as (x - 2)(x ^ 2 + 2x + 4)

3. Simplify the fraction by canceling common factors:

((x + 1)^2 / (x - 2)(x ^ 2 + 2x - 4)) ÷ ((x + 1) / (x - 2)(x ^ 2 + 2x + 4))

Cancel out the common factor of (x + 1) in the numerator and denominator:

((x + 1) / (x - 2)(x ^ 2 + 2x - 4)) ÷ (1 / (x - 2)(x ^ 2 + 2x + 4))

Invert the denominator and multiply:

((x + 1) / (x - 2)(x ^ 2 + 2x - 4)) * ((x - 2)(x ^ 2 + 2x + 4) / 1)

Cancel out common factors again:

(x + 1) * (x ^ 2 + 2x + 4) / (x - 2)

So the simplified form of the expression is:

(x + 1) * (x ^ 2 + 2x + 4) / (x - 2)