Question
What is the product in simplest form? Show all steps
(x+2)(x−2)/(x+1)(x−1)⋅(x+1)/x(x+2)
(x+2)(x−2)/(x+1)(x−1)⋅(x+1)/x(x+2)
Answers
GPT 3.5
To find the product of the given expression in simplest form, we need to first simplify each fraction and then multiply them together.
1. First, simplify each fraction separately:
(x+2)(x-2)/(x+1)(x-1) = [(x^2 - 4)/(x^2 - 1)] = [(x+2)(x-2)/(x+1)(x-1)]
(x+1)/x = 1 + 1/x
2. Now, multiply the two fractions together:
[(x+2)(x-2)/(x+1)(x-1)] * [(x+1)/x] = [(x+2)(x-2)(x+1)]/[(x+1)(x-1)x]
3. Simplify the expression by expanding and canceling out common terms:
[(x+2)(x-2)(x+1)]/[(x+1)(x-1)x]
= (x^2 - 4)(x+1)/(x^2 - 1)x
= (x^3 + x^2 - 4x - 4)/(x^3 - x)
= x(x^2 + x - 4 - 4/x)/(x^3 - x)
= (x^3 + x^2 - 4x - 4 - 4)/(x^3 - x)
= (x^3 + x^2 - 4x - 8)/(x^3 - x)
Therefore, the product of the given expression in simplest form is: (x^3 + x^2 - 4x - 8)/(x^3 - x)
1. First, simplify each fraction separately:
(x+2)(x-2)/(x+1)(x-1) = [(x^2 - 4)/(x^2 - 1)] = [(x+2)(x-2)/(x+1)(x-1)]
(x+1)/x = 1 + 1/x
2. Now, multiply the two fractions together:
[(x+2)(x-2)/(x+1)(x-1)] * [(x+1)/x] = [(x+2)(x-2)(x+1)]/[(x+1)(x-1)x]
3. Simplify the expression by expanding and canceling out common terms:
[(x+2)(x-2)(x+1)]/[(x+1)(x-1)x]
= (x^2 - 4)(x+1)/(x^2 - 1)x
= (x^3 + x^2 - 4x - 4)/(x^3 - x)
= x(x^2 + x - 4 - 4/x)/(x^3 - x)
= (x^3 + x^2 - 4x - 4 - 4)/(x^3 - x)
= (x^3 + x^2 - 4x - 8)/(x^3 - x)
Therefore, the product of the given expression in simplest form is: (x^3 + x^2 - 4x - 8)/(x^3 - x)