Question

How do you use the binomial series to expand `(x+2)^7`?

Answer 1

`x^7+14x^6+84x^5+280x^4+560x^3+672x^2+448x+128`

Explanation:

The seventh row of Pascal's triangle is

`1,7,21,35,35,21,7,1`

Thus,

`(a+b)^7=a^7+7a^6b+21a^5b^2+35a^4b^3+35a^3b^4+21a^2b^5+7ab^6+b^7`

This can be applied:

`(x+2)^7=x^7+7x^6(2)+21x^5(2)^2+35x^4(2)^3+35x^3(2)^4+21x^2(2)^5+7x(2)^6+(2)^7`

`=x^7+14x^6+84x^5+280x^4+560x^3+672x^2+448x+128`