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

1 Answer
Dec 19, 2015

x^7+14x^6+84x^5+280x^4+560x^3+672x^2+448x+128x7+14x6+84x5+280x4+560x3+672x2+448x+128

Explanation:

The seventh row of Pascal's triangle is

1,7,21,35,35,21,7,11,7,21,35,35,21,7,1

Thus,

(a+b)^7=a^7+7a^6b+21a^5b^2+35a^4b^3+35a^3b^4+21a^2b^5+7ab^6+b^7(a+b)7=a7+7a6b+21a5b2+35a4b3+35a3b4+21a2b5+7ab6+b7

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+2)7=x7+7x6(2)+21x5(2)2+35x4(2)3+35x3(2)4+21x2(2)5+7x(2)6+(2)7

=x^7+14x^6+84x^5+280x^4+560x^3+672x^2+448x+128=x7+14x6+84x5+280x4+560x3+672x2+448x+128