quick question on binomial therom

• Nov 6th 2008, 11:08 AM
johett
quick question on binomial therom
Find the general term, in simplified form...

\$\displaystyle (x^2-x)^10 \$ (exponent 10) and \$\displaystyle (a-1/a)^13\$ (exponent 13)

for the first one I got an answer of \$\displaystyle (10 r) x^20-r \$ (exponent 20) and the second one is \$\displaystyle (13 r) a^13-2r\$ (exponent 13) ....but what I can't get is the \$\displaystyle (-1)^r\$ included for both of the answers
• Nov 6th 2008, 12:00 PM
Laurent
Quote:

Originally Posted by johett
Find the general term, in simplified form...

\$\displaystyle (x^2-x)^{10} \$ and \$\displaystyle (a-1/a)^{13}\$

for the first one I got an answer of \$\displaystyle {10\choose r} x^{20-r} \$ and the second one is \$\displaystyle {13\choose r} a^{13-2r}\$ ....but what I can't get is the \$\displaystyle (-1)^r\$ included for both of the answers

One first thing: to write a long exponent, use curly brackets (like {10}) around it. You can click on the formulas in the corrected quote above to see how I changed it.

As for your question, the general formula is for \$\displaystyle (a+b)^n\$ but when you have \$\displaystyle (a-b)^n\$ (like here), this is \$\displaystyle (a+(-b))^n\$, hence the general term is \$\displaystyle {n\choose r}a^{n-r}(-b)^r=(-1)^ra^{n-r}b^r\$, hence the \$\displaystyle (-1)^r\$.
• Nov 6th 2008, 12:12 PM
johett
(Happy)(Happy)(Happy) *sigh* thanks for your help!! I was stressing over this cause I have a quiz tomorrow and I wasn't sure how to get the right answers.
And thanks for showing me how to properly get the exponents.