Question

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Answer 1

Step-by-step explanation:

The simplest possible form of f(n) is:

f(n) = n² + bn + c

f(n) / (n + 1) has a remainder of -12, so:

n² + bn + c = (n + p) (n + 1) − 12

n² + bn + c = n² + (1 + p)n + p − 12

b = 1 + p and c = p − 12

f(n) / (n + 2) has a remainder of -24, so:

n² + bn + c = (n + q) (n + 2) − 24

n² + bn + c = n² + (2 + q)n + 2q − 24

b = 2 + q and c = 2q − 24

Setting the expressions equal:

1 + p = 2 + q and p − 12 = 2q − 24

1 = p − q and p = 2q − 12

Solving the system of equations:

1 = q − 12

q = 13

p = 14

b = 15, c = 2

Therefore, f(n) is:

f(n) = n² + 15n + 2

Using grouping:

f(n) = n² + 3n + 2 + 12n

f(n) = (n + 1) (n + 2) + 12n

f(n) / ((n + 1) (n + 2) = 1 + (12n) / ((n + 1) (n + 2))

The remainder is 12n.