Explain this Algebra answer?

f(n) ≡ n3 + 7n2 + 14n + 3.

Express f(n) in the form f(n) ≡ (n + 1)(n + a)(n + b) + c,where a, b and c are integers.

Answer to this is: f(n) = (n + 1)(n + 2)(n + 4) − 5

However the next part asks: Hence, show that f(n) is odd for all positive integer values of n.

The answer is:

(n + 1) and (n + 2) are consecutive integers
∴ either (n + 1) or (n + 2) is even
∴ (n + 1)(n + 2)(n + 4) is even
∴ (n + 1)(n + 2)(n + 4) − 5 is odd

Can someone explain this?

which part are you unsure about?

I'm unsure about the answer to the second part. It doesn't make sense.

f(n) ≡ n3 + 7n2 + 14n + 3.

Express f(n) in the form f(n) ≡ (n + 1)(n + a)(n + b) + c,where a, b and c are integers.

Answer to this is: f(n) = (n + 1)(n + 2)(n + 4) − 5

However the next part asks: Hence, show that f(n) is odd for all positive integer values of n.

The answer is:

(n + 1) and (n + 2) are consecutive integers
∴ either (n + 1) or (n + 2) is even
∴ (n + 1)(n + 2)(n + 4) is even
∴ (n + 1)(n + 2)(n + 4) − 5 is odd

Can someone explain this?

As it says, n+1 and n+2 are two consecutive integers. Out of any pair of consecutive integers 1 must be even since every other integer is even.

Therefore one of n+1 or n+2 is divisible by 2. Hence (n+1)(n+2)(n+4) must be divisible by 2 hence that is even. An even number minus an odd number is odd hence (n+1)(n+2)(n+4) - 5 must be odd.

Not sure which part exactly you wanted explaining so I just did all of it.

Thanks! Wouldn't it be easier to write odd x even = odd therefore (n+1)(n+2)(n+4) is even. Even - odd = odd therefore (n+1)(n+2)(n+4) -5 is odd? How would I know which explanation I would get marks for?

Thanks! Wouldn't it be easier to write odd x even = odd therefore (n+1)(n+2)(n+4) is even. Even - odd = odd therefore (n+1)(n+2)(n+4) -5 is odd? How would I know which explanation I would get marks for?

You won't get any marks for writing "odd x even = odd"

But you should try to get into the habit of expressing yourself as mathematically as possible, so recognizing that n and n+1 are consecutive numbers and hence one must be even (and the equivalent statement for n+1 and n+2) is a useful phrase to know.

(Similarly, if you have something like n(n+1)(n+2) where n is an integer, you know that one factor must be divisible by 3 so the whole product is too.)

What are ways are there for expressing yourself as mathematically as possible? Do the exams just expect you to be able to do it?