Pascal's Triangle and The Binomial Theorem Examples
Example 1
Expand (x + y)3.
From Pascal's Triangle, we can see that our coefficients will be 1, 3, 3, and 1. The positive sign between the terms means that everything our expansion is positive.
(x + y)3
= 1x3 + 3x2y + 3xy2 + 1y3
= x3 + 3x2y + 3xy2 + y3
Notice that the sum of the exponents always adds up to the total exponent from the original binomial. Or don't. We're not the boss of you.
Example 2
Expand (a – 3b)4.
Pascal strikes again, letting us know that the coefficients for this expansion are 1, 4, 6, 4, and 1. The signs for each term are going to alternate, because of the negative sign.
(a – 3b)4
= 1a4 – 4a3(3b) + 6a2(3b)2 – 4a(3b)3 + 1(3b)4
Be sure to put all of 3b in the parentheses. It'd be a shame to leave that 3 all on its lonesome.
= a4 – 12a3b + 6a2(9b2) – 4a(27b3) + 81b4
= a4 – 12a3b + 54a2b2 – 108ab3 + 81b4
Depending on what the terms look like inside the binomial, the end result can look very different from what Pascal initially tells us.
Example 3
Expand (2x + y)5.
The mighty Triangle has spoken. The coefficients are 1, 5, 10, 10, 5, and 1.
A binomial raised to the 6th power is right around the edge of what's easy to work with using Pascal's Triangle. You can go higher, as much as you want to, but it starts to become a chore around this point.
