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# Number Theory/find factors

Question

The first thing to do is factor out the 3.
This gives 3(x² + 4x - 837).

Next, note that the 837 is divisible 3, giving 3*279; 279 = 3*93; 93 = 3*31.
That means 837 = 3*3*3*31.  Since 3*3*3 = 27, and that is 4 less than 31,
the factors to use are +31 and -27.  For x=31, x-31=0.  For x=-27, x+27=0.

This finally makes it be 3(x-31)(x+27).

To do a different approach ...

If you recognize the squares, 837 = 841 - 4, where 841 is 29² and 4 is 2².
This means the problem will have factors involving 29+2 and 29-2.
Since those are 31 and 27, and the middle term is a positive (+4),
add the 31 and subtract the 27, gives 3(x-31)(x+27).

Number Theory

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