Patrick W. answered • 04/13/15
Tutor
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High School Mathematics Teacher, Passionate Math Geek
You obviously don't want to expand this all out, but we can get some help from Pascal's Triangle
First, you've correctly labeled our polynomial as a binomial. Instead of (x+2p÷x), I'm going to write it as (x+2px-1).
Second, the nth row of Pascal's Triangle gives us the correct coefficients of the expanded binomial (a+b)n
Google Pascal's Triangle and check the eighth row. Here's what I found:
1 8 28 56 70 56 28 8 1
These will be the coefficients of our terms, listed in decreasing order of 'a' terms and increasing order of 'b' terms.
1(x)8(2px-1)0
+8(x)7(2px-1)1
+28(x)6(2px-1)2
+56(x)5(2px-1)3
+70(x)4(2px-1)4
+56(x)3(2px-1)5
+28(x)2(2px-1)6
+8(x)1(2px-1)7
+1(x)0(2px-1)8
All of these terms can be simplified. I hope you'll see now why I chose to write my 1/x terms in a way that would help me keep track of cancellations:
(x)8+8(x)6(2p)+28(x)4(2p)2+56(x)2(2p)3+70(2p)4+56(x)-2(2p)5+28(x)-4(2p)6+8(x)-6(2p)7+(x)-8(2p)8
There is one term with x6 in it, the second term 8(x)6(2p) which I'll simplify as 16px6
I hope that helps! Comment back if any of that didn't connect for you.
