How do you use pascals triangle to expand (d - 5)^6(d−5)6?
1 Answer
Combine the row of Pascal's triangle that starts with
(d-5)^6 = d^6-30d^5+375d^4-2500d^3+9375d^2-18750d+15625(d−5)6=d6−30d5+375d4−2500d3+9375d2−18750d+15625
Explanation:
Write out Pascal's triangle as far as the row that begins
Write out this row on its own:
11 ,66 ,1515 ,2020 ,1515 ,66 ,11
Write out ascending powers of
11 ,-5−5 ,2525 ,-125−125 ,625625 ,-3125−3125 ,1562515625
Multiply these two sequences together:
11 ,-30−30 ,375375 ,-2500−2500 ,93759375 ,-18750−18750 ,1562515625
These are the coefficients we need for our binomial expansion:
(d-5)^6 = d^6-30d^5+375d^4-2500d^3+9375d^2-18750d+15625(d−5)6=d6−30d5+375d4−2500d3+9375d2−18750d+15625
In general:
(a+b)^n = sum_(k=0)^n ((n), (k)) a^(n-k) b^k
where
If you look at the way Pascal's triangle is constructed, with each number being the sum of the two numbers above it, you find that each number is equal to the number of ways that you can reach it starting from the top of the triangle and stepping to the left or right on your way down.
This corresponds to the number of ways that you can choose left and right terms to get a particular power of
(a+b)^n = stackrel "n times" overbrace((a+b)(a+b)...(a+b))
