Does anyone know of an easy to remember way of expanding binomials, I do have a grasp of it but wondering if theres a better way of doing it?
An example would be (e + f)^5
apparently you know the Pascal's triangle method, here's how to implement it.
we have the Pascal's triangle:
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
these numbers give the coefficients for all the terms in a binomial expansion. so we place them into position as they are in the triangle. then we take the products of the two terms in the binomials and place them in the following order.
the first power of the first element is the highest power of the expansion, we start at the highest power and count down by 1 until we get to zero. for the second element, we start with power zero and count up till we get to the highest power.
so for say (a + b)^n the terms will have the form, Ca^(n - k)*b^(k), where C is the coefficient given by Pascal's triangle and k = 0,1,2,3...n
so for instance:
(x + 2)^4 = 1*x^4(2)^0 + 4*(x^3)(2^1) + 6*(x^2)(2^2) + 4*(x^1)(2^3) + 1*(x^0)(2^4)
notice that the numbers in front are given by the line of Pascal's triangle where the second number is 4, and that i start with the power of x (the first element) as 4 and count down, and start with the power of 2 (the second element) as 0 and count up
so, (x + 4)^4 = 1x^4 + 4*2*x^3 + 6*4*x^2 + 4*8*x + 1*16
...................= x^4 + 8x^3 + 24x^2 + 32x + 16
do you get it now?
now try your question again and tell me your answer
As you see from Plato's post, you have to know what factorials are to use the Binomial expansion method, you can do it that way as well...you do know what factorials are right?
Both the methods are easy to learn and easy to implement and i use both depending on what i want. Pascal's triangle i use to expand smaller binomial expansions, say where the highest power is like 6,7, or maybe 8. i use the binomial expansion method when dealing with higher powers
one advantage of the Binomial expansion method however, is that you can tell what any term is right of the bat.
say for instance we ONLY want the 17th term of (x + 6)^28
with Pascal's triangle method you would need to more or less calculate all the coefficients before that, not to mention draw Pascal's triangle to line 17. with the binomial expansion method, the answer is immediate, as Plato illustrated