Difference between revisions of "AoPS Wiki talk:Problem of the Day/July 1, 2011"
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==Problem== | ==Problem== | ||
{{:AoPSWiki:Problem of the Day/July 1, 2011}} | {{:AoPSWiki:Problem of the Day/July 1, 2011}} | ||
− | ==Solution== | + | ==Solution 1== |
− | {{ | + | To factor <math>(n+1)(n+2)(n+3)(n+4) - 120</math>, we should try to find a way to create a quadratic in disguise. There, in fact, is a way! |
+ | |||
+ | Expand <math>(n+1)(n+4)</math> and <math>(n+2)(n+3)</math> separately: | ||
+ | |||
+ | <math>(n+1)(n+4)(n+2)(n+3) = (n^2+5n+4)(n^2+5n+6)</math> | ||
+ | |||
+ | We notice that there is a <math>n^2 + 5n</math> in both of these terms! Treat <math>n^2 + 5n</math> as a single quantity, <math>x</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>. | ||
+ | |||
+ | <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>. | ||
+ | ==Solution 2== | ||
+ | Let <math>P(x) = (x+1)(x+2)(x+3)(x+4) - 120</math>. Then <math>P(1) = 0</math> and <math>P(-6)=0</math> (since <math>120 = -5\cdot -4\cdot -3\cdot -2</math>). Therefore I have: | ||
+ | <cmath>P(x) = (x-1)(x+6)Q(x)</cmath> | ||
+ | where <math>Q</math> is a quadratic. Simplifying yields <math>Q(x)=x^2+5x+16</math> as before, which is irreducible as <math>16 > \frac{5^2}{4}</math>. |
Latest revision as of 19:20, 1 July 2011
Problem
AoPSWiki:Problem of the Day/July 1, 2011
Solution 1
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 .
can be factored as .
Our final factorization is .
Solution 2
Let . Then and (since ). Therefore I have: where is a quadratic. Simplifying yields as before, which is irreducible as .