Newton's Binomial Example

The Newton's binomial, also called "binomial theorem " is a logarithm that allows us to obtain powers of binomials.

To obtain the binomial power, the coefficients called “binomial coefficients"Which consist of sequences of combinations.

Example 1, General formulas of Newton's binomial:

(a + b)² = a² + 2 ab + b²
(a - b)² = a² –2 ab + b²
(a + b) 3 a³ + 3 to²b + 3 ab² + b³

These formulas are known by the name of notable identities, where a more general formula is created that is equivalent to the development of (a + b)ⁿ, where n is any natural integer.

This formula is valid for any element to Y b of a ring,

A (for laws + Y x) to

Condition that the two elements toY b be such that to x b = b x to:

(a + b)ⁿ = aⁿ + C¹_n to^n-2 xb² + ...
+ C^p_n  to^n-p x b^p +… + C_pⁿ¹ + bⁿ.

The C^p_n are natural integers, called binomial coefficients (those that express the number of combinations of n items taken p to p; can be easily calculated thanks to Pascal's triangle).

Example 2, from Newton's binomial:

We consider multiplication:

z. z = z² where z can be any algebraic expression:

Now suppose that z = x + Y, then:

z. z = (x + y) = (x + y) but (x + y)

which can be calculated like this:

x + y
x + y

Here the multiplication is carried out from left to right and the result is obtained by adding algebraically:

x² + x y
+ xy + y²

x² + 2 x y + y²

(x + y)² = x² + 2 x y + y²

If we consider:

z. z. z = z³;

(x + y) (x + y) (x + y) = (x + y)². (x + y) 2. (x + y) = (x² + 2 xy + y²) (x + y)

When the multiplication is carried out we obtain:

X2 + 2 x y + y2
+ x²y + 2 x y² + and²

X³ + 3 x² y + 3 x y² + and³

(x + y)² (x + y) = (x + y)³ = x³ + 3 x² y + 3 x y² + and³.

 z³. z = z⁴

z3. z = (x3 + 3 x2 y + 3 x y2 + y3) (x + y)

And when we do the multiplication.

x³ + x² y + 3 x y² + and³

x + y_________________

x⁴ + 3 x³ y + 3 x² Y² + x y³

+ x³ y + 3 x2 y2 + 3xy³ + and⁴

x⁴ + 4x³and + 6x² y + 4xy³ + and⁴

(x + y)⁴ = x4 + 4x³and + 6x² Y² + 4xy³ + and⁴