Question

What is coefficient of the `x^3` term in the binomial expansion of `(4 - x)^9`?

Answer 1

`-344064x^3`

Explanation:

There's a quick way to answering questions like this without the need to go through the whole process of finding the terms before the `x^3` term:

Using the `color(white)()^n C_r` button, the value of `n` represents the expansion power of `9` and the `r` would be the term you want to find, so in this case, it would be `3`.

Therefore, the coefficient of the general `x^3` term is:

`color(white)()^9 C_3= 84`

However, you can only do this if the expansion is in the format of `(1+x)^n`.

To find the `x^3` coefficient for `(4-x)^9`:

`(a+x)^n=a^n+na^(n-1)x+((n(n-1))/(2!))a^(n-2)x^2+((n(n-1)(n-2))/(3!))a^(n-3)x^3+......+x^n`

Take just the `x^3` section of the binomial theorem expansion equation:

`((n(n-1)(n-2))/(3!))a^(n-3)x^3` and substitute values of `n`, `x` and `a` in where `n=9` and `x=-1` and `a=-4`:

`((9(9-1)(9-2))/(3!))(-4)^(9-3)(-1)^3 = -344064`

Therefore, the `x^3` term for the binomial expansion of `(4-x)^9` is `-344064x^3`

(Note: Factorial '!' is a product of an integer and all the integers below it. For example, `(4!)=4*3*2*1`