divide ((x^2+2x+1)/x^2-8x+16))/((x+1/x^2-16))
To divide the given rational expression, we first factor the polynomials in the numerator and denominator:
Numerator: x^2 + 2x + 1 = (x+1)(x+1)
Denominator: x^2 - 8x + 16 = (x-4)(x-4); x + 1 = (x^2 + 1)/x^2; x^2 - 16 = (x+4)(x-4)
So the given rational expression can be written as:
[(x+1)(x+1)/(x-4)(x-4)] / [(x^2+1)/x^2 - (x+4)(x-4)/(x-4)(x-4)]
To simplify this expression, we first find a common denominator:
[(x+1)(x+1)/(x-4)(x-4)] / [(x^2+1)(x-4)^2/x^2 - (x+4)(x-4)/(x-4)(x-4)]
Now we can simplify by multiplying the first fraction by x^2/x^2:
[(x+1)(x+1)x^2/(x-4)(x-4)x^2] / [(x^2+1)(x-4)^2 - (x+4)(x-4)x^2/(x-4)(x-4)x^2]
Simplifying further with the distributive property:
[(x^3 + 2x^2 + x)/(x-4)^2x^2] / [(x^4 - 8x^2 + 16) - (x^3 - 4x^2 + 4x - 16)]/(x-4)^2x^2
Simplifying the second fraction by combining like terms:
[(x^3 + 2x^2 + x)/(x-4)^2x^2] / [(x^4 - x^3 - 4x^2 + 4x)/ (x-4)^2x^2]
Now we can divide by multiplying the numerator of the first fraction by the reciprocal of the second fraction:
[(x^3 + 2x^2 + x)/(x-4)^2x^2] * [(x-4)^2x^2/(x^4 - x^3 - 4x^2 + 4x)]
Simplifying by canceling out the common factors:
(x^3 + 2x^2 + x)/(x^4 - x^3 - 4x^2 + 4x - 16)
This is the simplified form of the given rational expression.