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2000 SMT/Algebra Problems/Problem 2

Revision as of 14:14, 18 April 2023 by Binoutheguineapig (talk | contribs) (→‎Solution 1 - SUBMITTED BY HOWDOI_YT)
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Problem 2

Evaluate $2000^3 - 1999(2000^2) - 1999^2(2000) + 1999^3$

Solution 1 - Submitted by howdoi_yt

I can rewrite the first $1999$ as $(2000-1)$, and the third $2000$ as $(1999+1)$;

$2000^3 - 2000^2(2000-1) - 1999^2(1999+1) + 1999^3$

$= 2000^3 - 2000^3 + 2000^2 - 1999^3 - 1999^2 + 1999^3$

$= 2000^2 - 1999^2$

$= (2000 - 1999)(2000 + 1999)$

$= 3999 \qquad \blacksquare$

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