((x ^ 2 + 2x + 1)/(x ^ 3 - 8x + 16))/((x + 1)/(x ^ 3 - 16))
To simplify the expression ((x^2 + 2x + 1)/(x^3 - 8x + 16))/((x + 1)/(x^3 - 16)), we can start by simplifying each fraction individually and then dividing them. Let's break it down step-by-step:
Step 1: Simplify the numerator of the first fraction, (x^2 + 2x + 1).
The numerator, x^2 + 2x + 1, is already in its simplest form and cannot be factored further.
Step 2: Simplify the denominator of the first fraction, (x^3 - 8x + 16).
The denominator, x^3 - 8x + 16, can be factored as (x - 2)(x^2 + 2x - 8).
Step 3: Simplify the numerator of the second fraction, (x + 1).
The numerator of the second fraction, x + 1, is also in its simplest form and cannot be simplified further.
Step 4: Simplify the denominator of the second fraction, (x^3 - 16).
The denominator, x^3 - 16, can be factored using the difference of cubes formula as (x - 2)(x^2 + 2x + 4).
Step 5: Apply the division of fractions rule.
Dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. Therefore, we can rewrite the expression as ((x^2 + 2x + 1)(x^2 + 2x + 4))/((x^3 - 8x + 16)(x + 1)) * ((x + 1)(x^2 + 2x - 8))/((x - 2)(x^2 + 2x + 4)).
Step 6: Simplify the expression.
Many terms in the numerator and denominator will cancel each other out. We can cancel the common factors (x^2 + 2x + 4) and (x + 1) in both the numerator and denominator.
After canceling out, the simplified expression is:
(x^2 + 2x + 1)(x^2 + 2x - 8)/(x^3 - 8x + 16)(x - 2)
This is the simplified form of ((x^2 + 2x + 1)/(x^3 - 8x + 16))/((x + 1)/(x^3 - 16)).
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)