Question

What is the third term in the expansion of the binomial `(2x+5)^5`?

Answer 1

2000`x^3`

Explanation:

All binomials to powers can be written as `(a+b)^x`
While expanded, two rules come forward.

1. The result can be written as the addition of all powers of a times all powers of b, where the power on the a term plus the power on the b term is equal to x.
   For example, `(a+b)^2` can be written as `a^2+2ab+b^2`. On the `a^2` term, the power a is being raised to is 2, and the power that b is being raised to is 0. `b^0` is equal to 1, and `1*a^2=a^2`, so the term is `a^2`. This will work for the other terms as well

2.There are coefficients on the terms, that increase as you reach the middle. For the sake of a (fairly) short answer, I'll assume you know Pascal's triangle.

If you know Pascal's triangle, just assign each row as being the coefficients when (a+b) is raised to the power of x-1. For example, the 4th row of Pascal's Triangle produces the result of 1 3 3 1, so if you raise (a+b) to the 4-1, or third power (equivalent to `(a+b)^3`), you get the coefficients of 1 3 3 1. You then assign `a^3`to the first coefficient, `a^2b` to the second, `ab^2`to the third, and `b^3`to the fourth, and then add them up, ending up with `a^3+3a^2b+3b^2a+b^3`.

Now, for your problem input 2x as a, 5 as b, and 5 as x. We are only looking for the third term, so we go to the value where a is only being raised to the 3rd power. As a result, b must be raised to the second power, so the powers on a and b are equal to 5. This produces the result of `(2x)^3*5^2=8x^3*25=200x^3`. Finally, using Pascal's Triangle, we know the third term is 10, so we multiply that by the amount already there, obtaining the result of `10*200x^3=2000x^3` which is our answer