Question

What is the coefficient of x in the expansion of (x-3)^7?

Answer 1

`5103`

Explanation:

I don't know if you have learnt about Pascal's triangle,
and how to expand a binomial from it


We can for example expand binomial, (possible see more here)

`(x+y)^0=1`

`(x+y)^1=x+y`

`(x+y)^2=x^2+2xy+y^2`

`(x+y)^3=x^3+3x^2y+3xy^2+y^3`

And so on
Notice how the binomial expansion depends of the lines of pascal's triangle,

We can also see the term involving `x` is always the second last

We look for

`(x+y)^7=x^7+7x^6y+...+color(red)(7xy^6)+y^7`

We only care about `7xy^6`, cause it's the only term involving `x`
In your example `y=-3`, thus

`color(blue)(7(-3)^6)x=color(blue)(7*729)x=color(blue)(5103)x`

The coefficient of `x` is `5103`