Binomial expansion

Binomial expansion uses an expression to make a series. It uses a bracket expression like $(x + y)^{n}$ . There are three binomial expansions.

The formulas

There are basically three binomial expansion formulas:

We can explain why there are such 3 formulas with a simple expansion of the product:

(a + b)^{2} = (a + b) \cdot (a + b) = a \cdot a + a \cdot b + b \cdot a + b \cdot b = a^{2} + 2 \cdot a \cdot b + b^{2}

(a - b)^{2} = (a - b) \cdot (a - b) = a \cdot a - a \cdot b - b \cdot a + b \cdot b = a^{2} - 2 \cdot a \cdot b + b^{2}

(a + b) \cdot (a - b) = a \cdot a - a \cdot b + b \cdot a - b \cdot b = a^{2} - b^{2}

If $n$ is an integer ( $n \in ℤ$ ), we use Pascal's triangle.

To expand $(x + y)^{2}$ :

find row 2 of Pascal's triangle (1, 2, 1)
expand $x$ and $y$ so the $x$ power goes down by 1 each time from $n$ to 0 and the $y$ power goes up by 1 each time from 0 up to $n$
times the numbers from Pascal's triangle with the right terms.

So $(x + y)^{2} = 1 x^{2} y^{0} + 2 x^{1} y^{1} + 1 x^{0} y^{2}$

For example:

(3 + 2 x)^{2} = 1 \cdot 3^{2} \cdot (2 x)^{0} + 2 \cdot 3^{1} \cdot (2 x)^{1} + 1 \cdot 3^{0} \cdot (2 x)^{2} = 9 + 12 x + 4 x^{2}

So as a rule:

(x + y)^{n} = a_{0} x^{n} y^{0} + a_{1} x^{n - 1} y^{1} + a_{2} x^{n - 2} y^{2} + \dots + a_{n - 1} x^{1} y^{n - 1} + a_{n} x^{0} y^{n}

where $a_{i}$ is the number at row $n$ and position $i$ in Pascal's triangle.

(5 + 3 x)^{3} = 1 \cdot 5^{3} \cdot (3 x)^{0} + 3 \cdot 5^{2} \cdot (3 x)^{1} + 3 \cdot 5^{1} \cdot (3 x)^{2} + 1 \cdot 5^{0} \cdot (3 x)^{3}

= 125 + 75 \cdot 3 x + 15 \cdot 9 x^{2} + 1 \cdot 27 x^{3} = 125 + 225 x + 135 x^{2} + 27 x^{3}

(5 - 3 x)^{3} = 1 \cdot 5^{3} \cdot (- 3 x)^{0} + 3 \cdot 5^{2} \cdot (- 3 x)^{1} + 3 \cdot 5^{1} \cdot (- 3 x)^{2} + 1 \cdot 5^{0} \cdot (- 3 x)^{3}

= 125 + 75 \cdot (- 3 x) + 15 \cdot 9 x^{2} + 1 \cdot (- 27 x^{3}) = 125 - 223 x + 135 x^{2} - 27 x^{3}

(7 + 4 x^{2})^{5} = 1 \cdot 7^{5} \cdot (4 x^{2})^{0} + 5 \cdot 7^{4} \cdot (4 x^{2})^{1} + 10 \cdot 7^{3} \cdot (4 x^{2})^{2} + 10 \cdot 7^{2} \cdot (4 x^{2})^{3} + 5 \cdot 7^{1} \cdot (4 x^{2})^{4} + 1 \cdot 7^{0} \cdot (4 x^{2})^{5}

= 16807 + 12005 \cdot 4 x^{2} + 3430 \cdot 16 x^{4} + 490 \cdot 64 x^{6} + 35 \cdot 256 x^{8} + 1 \cdot 1024 x^{10}

= 16807 + 48020 x^{2} + 54880 x^{4} + 31360 x^{6} + 8960 x^{8} + 1024 x^{10}