Most students who take algebra learn about FOIL, which stands for First, Outer, Inner, Last. It tells the order of multiplication for a binomial like (a + b).
(a + b)^2 = (a + b)(a +b) = a^2 + ab + ba + b^2 = a^2 + 2ab +b^2
It's easy enough for (a + b)^2, but what about (a + b)^3 or a greater exponent?
A mathematician named Pascal noticed a pattern when expanding binomials. The coefficients can be arranged into a triangle, like so:
1
1 2 1
1 3 3 1
1 4 6 4 1
Do you see the pattern? Take a few minutes to work out the next row. Hint: the first and last numbers will be 1 and the rest follow a rule from the row above.
The next row is 1 5 10 10 5 1, and those are the coefficients for the expansion of (a + b)^5. The 1s and 4s in the previous row make 5 in two places and the two 4s and the 6 make 10 in two places. Filling out this chart is much easier than trying to work this out with FOIL. Let's do that anyway just to double check.
(a + b)^5 = (a + b)(a + b)^4 =
(a + b)(a + b)^2(a + b)^2 =
(a + b)(a^2 + 2ab + b^2)(a^2 + 2ab + b^2) =
(a + b)(a^2 + 2ab + b^2)^2 =
(a + b)(a^4 + 2a^3b + a^2b^2 + 2a^3b + 4a^2b^2 + 2ab^3 + a^2b^2 + 2ab^3 + b^4) =
(a + b)(a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4) =
(a^5 + 4a^4b + 6a^3b^2 + 4a^2b^3 + ab^4 + a^4b + 4a^3b^2 + 6a^2b^3 + 4ab^4 + b^5) =
a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
Whew! That's a lot of algebra. It's much easier just to use Pascal's Triangle.
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