Question

How do you expand (c-d)^7 ?

Answer 1

`c^7-7c^6d+21c^5d^2-35c^4d^3+35c^3d^4-21c^2d^5+7cd^6-d^7`

Explanation:

Use Pascal's Triangle to expand. We use Pascal's triangle to find the coefficients. The level you go to is determined by the power. So, we go to the 7th level to find the coefficients for the expansion.

However, know that Pascal's Triangle's first row is called row 0!

Pascal's Triangle:

So, using the seventh row, the coefficients are:
`1, 7, 21, 35, 35, 21, 7, 1`

Now, add the terms. To do this, you take the first term in the binomial and plug it in after each coefficient, with the power decreasing by 1 each time. It looks like this:

`1*(c)^7+7*(c)^6+21*(c)^5+35*(c)^4+35*(c)^3+21*(c)^2+7*(c)^1+1*(c)^0`

Now, you do the same thing for the second term in the binomial, but for the second term, start with the power at 0 and increase by 1 each time. Also, notice that the second term is negative! So, it looks like this:

`1*(c)^7*(-d)^0+7*(c)^6*(-d)^1+21*(c)^5*(-d)^2+35*(c)^4*(-d)^3+35*(c)^3*(-d)^4+21*(c)^2*(-d)^5+7*(c)^1*(-d)^6+1*(c)^0*(-d)^7`

Next, all you have to do is simplify based on exponents.

`1*c^7*1+7*c^6*-d+21*c^5*d^2+35*c^4*-d^3+35*c^3*d^4+21*c^2*-d^5+7*c*d^6+1*1*-d^7`

And finally, just simplify.

`c^7-7c^6d+21c^5d^2-35c^4d^3+35c^3d^4-21c^2d^5+7cd^6-d^7`