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