Algebraic expansions to memorize

Key algebraic expansions include:

(a^2-b^2)= (a+b)(a-b)

(a+b)^2= a^2 +2ab+ b^2

(a-b)^2= a^2 -2ab+ b^2

(a+b)^3= (a+b)(a^2 -ab +b^2)

(a-b)^3= (a-b)(a^2 +ab +b^2)

The above algebraic expansions will be helpful for both MathCounts, factoring equations, and math class. Comment below if you have anything to add.

If you take A+B^3 and you want to find out the answer, then imagine pascal's triangle. If you want to find out what a+b^5 is then find the fifth line in pascal's triangle which is 1 5 10 10 5 1. These are the coefficients. As for the variables, the first would be a^5 then a^4b then a^4b^3 and so on until the last one which should be b^5. This will help so much with FOIL-related problems.

theonlyMattinMathCounts wrote:If you take A+B^3 and you want to find out the answer, then imagine pascal's triangle. If you want to find out what a+b^5 is then find the fifth line in pascal's triangle which is 1 5 10 10 5 1. These are the coefficients. As for the variables, the first would be a^5 then a^4b then a^4b^3 and so on until the last one which should be b^5. This will help so much with FOIL-related problems.

Yup. For those people who enjoy collecting theorems, this is known as the Binomial Theorem.

This can prove very efficient for long arithmetic problems as in:
Determine (5^3) + 3(5^2)(2) + 3(5)(4) + 2^3

Spoiler:

5 second problem. Note that it is (a+b)^3, where a and b are 5 and 2. (5+2)^3= 343.