Question

How do you expand `(x+2)^7`?

Answer 1

`x^7+14x^6+84x^5+280x^4+560x^3+672x^2+448x+128` using the binomial theorem.

Explanation:

The binomial theorem is the theorem that allows you to do this. It's kinda based off of Pascal's triangle and an easy little "gimmick" as it were.

Pascal's triangle is a nifty little geometric arithmetic device, allow me to use this picture to explain

As you can see a line of 1's goes down the left and right, this is to ensure it keeps going. Essentially once you have two numbers next to each other, their sum will be the number directly below the middle of them.

Now how this relates to the binomial theorem is that a certain row on Pascal's triangle will be the base coefficients for your terms. Now the rest of binomial theorem is basically the coefficient * term `a` to the highest power but the power will decrease by one for each additional term * term `b` to the 0th power but will add one to it each time there's another term. And you will then add all these terms up. Perhaps this is best illustrated with a small example.

`(x+2)^3` will have 4 terms total. Term `a` would be `x` and term `b` would be `2`. Since it has four terms we need the fourth row of Pascal's triangle which is `1,3,3,1`. So now the order of arithmetic is as follows.

`1*x^3*2^0 + 3*x^2*2^1+3*x^1*2^2+1*x^0*2^3`

`x^3+6x^2+12x+8` will be our final answer, now we have to do this for your equation.

`(x+2)^7` will have 8 terms so we must use the 8th row of Pascal's triangle which is `1,7,21,35,35,21,7,1`

`1*x^7*2^0+7*x^6*2^1+21*x^5*2^2+35*x^4*2^3+35*x^3+2^4+21*x^2*2^5+7*x^1*2^6+1*x^0*2^7`

Whew that's a monster, well after you work out all of that you get your answer `x^7+14x^6+84x^5+280x^4+560x^3+672x^2+448x+128`