What is the difference in simplest form?
n^2+3n+2/n^2+6n+8 - 2n/n+4
To find the difference in simplest form, we need to find a common denominator for both fractions and then combine them.
The first fraction is n^2 + 3n + 2 / n^2 + 6n + 8.
Factoring the numerator and denominator, we have:
n^2 + 3n + 2 = (n + 1)(n + 2)
n^2 + 6n + 8 = (n + 2)(n + 4)
Therefore, the first fraction simplifies to:
(n + 1)(n + 2) / (n + 2)(n + 4)
The second fraction is 2n / n + 4.
To make the denominator of the first fraction the same as the denominator of the second fraction, we need to multiply the first fraction by (n + 4) / (n + 4).
Therefore, the first fraction becomes:
(n + 1)(n + 2) * (n + 4) / (n + 2)(n + 4)
Combining the two fractions, we get:
((n + 1)(n + 2)(n + 4) - 2n) / (n + 2)(n + 4)
Expanding the numerator we get:
(n^2 + 6n + 8)(n + 1) - 2n = n^3 + 7n^2 + 14n + 8 - 2n = n^3 + 7n^2 + 12n + 8,
so the difference in simplest form is (n^3 + 7n^2 + 12n + 8) / (n + 2)(n + 4).