Expansion

Focus Question: How can the triangle be used in modern mathematics?
(1m+a)^5

=(1m+a)(1m+a)(1m+a)(1m+a)(1m+a)

=1m^5+5(〖m)〗^4 (a)^1+10(〖m)〗^3 (a)^2+10(〖m)〗^2 (a)^3+5(m)^1 (〖a)〗^4+1(〖a)〗^5

The coefficients of the the pro numerals are the same as the numbers of the pascals triangle. The exponent at the top is referring to the row on pascals triangle excluding the first one. This example is referring to the 5th row. 1 - 5 - 10 - 10 - 5 - 1

Here is another example for the 10th row of Pascal's Triange:

(1m+a)^10

=(1m+a)(1m+a)(1m+a)(1m+a)(1m+a)(1m+a)(1m+a)(1m+a)(1m+a)(1m+a)

=1m^10+10m^9 a^1+45m^8 a^2+120m^7 a^3+210m^6 a^4+252m^5 a^5+210m^4 a^6+120m^3 a^7+45m^2 a^8+10m^1 a^9+,〖1a〗^10



F.O.I.L
A very important rule that Pascals triangle relates to is the expanding rule F.O.I.L (First, Outer, Inner, Last). FOIL is a rule used to solve binomial expansions. The FOIL rule tells us to multiply the first two numbers in each bracket, the outer numbers in each bracket, the inner numbers in each bracket and also the last numbers in each bracket. According to what signs you have, you add or subtract these numbers to get your final result. The example below shows how FOIL works.