Binomial Theorem

Binomial= a polygon that has two terms.

Expanding a Binomial:

(x + y)¹ = 1x + 1y

(x + y)² = 1x² + 2xy + 1y²

(x + y)³ = 1x³ + 3x²y + 3xy² +1y³

(x + y)⁴ = 1x⁴ + 4x³y + 6x²y² + 4xy³ + 1y⁴

In each expansion there is (n + 1) terms

n= 2 → (2+1) = 3 terms

n= 3 → (3+1) = 4 terms

The power of each term adds up to the power each binomial is added to (n).

For, (x + y)⁴ = 1x⁴ + 4x³y¹ + 6x²y² + 4x¹y³ + 1y⁴

Powers: (4) (4) (3+1= 4) (2+2=4) (1+3=4) (4)

where n = 4, the sum of powers are 4.

Coefficients of the expansion correspond to the Pascal's Triangle

*During expansion, the x and y have symmetric roles. As the powers of "x" decrease by 1, the powers of "y" increase by 1.

(x + y)⁴ = x⁴(y⁰) + 4x³y¹ + 6x²y² + 4x¹y³ + (x⁰)y⁴

To find the coefficents you can also use the formula:

_nC_{k =}

The 2 values next to each others' sum is equal to the number directly below the values. This is the case for all the numbers on Pascal's triangle.