Expand the expression by using the Pascal's Triangle and the Binomial Theorem.
x^6+6x^5+15x^4+20x^3+15x^2+6x+1
Practice makes perfect
To expand the binomial, we should recall the Binomial Theorem. It states that for every positive integer n, we can expand the expression (a+b)^n by using the numbers in the n^(th) row of Pascal's Triangle.
(a+b)^n
=
P_0a^nb^0+P_1a^(n-1)b^1+... +P_(n-1)a^1b^(n-1)+P_na^0b^n
In the above formula, P_0, P_1, ..., P_n are the numbers in the n^(th) row of Pascal's Triangle.
l&l&l&l&l&l&l&l&l&l&l&l&l&l
Row&&&&&&Pascal's&&&&&&&
&&&&&&Triangle&&&&&&&
c&c&c&c&c&c&c&c&c&c&c&c
0& & & & & & &1 & & & & & &
1& & & & & &1 & &1 & & & & &
2& & & & &1 & &2 & &1 & & & &
3& & & &1 & &3 & &3 & &1 & & &
4& & &1 & &4 & &6 & &4 & &1 & &
5 & &1 & &5 & &10 & &10 & &5 & &1
6 & 1 & & 6 & & 15 & & 20 & & 15 & & 6 & & 1
Note that each number found in the triangle that is the sum of the two numbers diagonally above it. Now consider the given binomial.
( x+ 1 )^6
We can substitute the first term for a and the second term for b using the Binomial Theorem equation and the coefficients from Pascal's Triangle.
