Difference between revisions of "AoPS Wiki talk:Problem of the Day/July 1, 2011"
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Our original quantity was <math>(x + 4)(x+6)-120 = x^2+10x+24-120=x^2+10x-96=(x-6)(x+16)</math>. | Our original quantity was <math>(x + 4)(x+6)-120 = x^2+10x+24-120=x^2+10x-96=(x-6)(x+16)</math>. | ||
− | We can factor the original polynomial as <math>[(n^2+5n)-6][(n^2+5n)+16] = (n^2+5n-6)(n^2+5n+16)</math> | + | We can factor the original polynomial as <math>[(n^2+5n)-6][(n^2+5n)+16] = (n^2+5n-6)(n^2+5n+16)</math>. |
Notice that <math>n^2+5n-6</math> can be factored as <math>(n-1)(n+6)</math>. | Notice that <math>n^2+5n-6</math> can be factored as <math>(n-1)(n+6)</math>. | ||
Our final factorization is <math>\boxed{(n-1)(n+6)(n^2+5n+16)}</math>. | Our final factorization is <math>\boxed{(n-1)(n+6)(n^2+5n+16)}</math>. |
Revision as of 20:22, 30 June 2011
Problem
AoPSWiki:Problem of the Day/July 1, 2011
Solution
To factor , we should try to find a way to create a quadratic in disguise. There, in fact, is a way!
Expand and separately:
We notice that there is a in both of these terms! Treat as a single quantity, .
Our original quantity was .
We can factor the original polynomial as .
Notice that can be factored as .
Our final factorization is .