A-4: Algebraic Expressions and Operations
From first principles, expand x+y5 by the Binomial theorem.
The Binomial theorem: x+yn=∑k=0nnkxn−k yk, where nk=n!k! n−k! is the binomial coefficient, and n! =n⋅n−1⋅ ⋯ ⋅1 is read "n factorial." By definition, 0!=1.
=∑k=055k x5−k yk
=5!0! 5!x5+5!1! 4!x4y+5!2! 3!x3y2+5!3! 2!x2y3+5!4! 1!xy4+5!5! 0!y5
Expression palette: Summation template ∑i=knf and binomial coefficient ab template
Context Panel: Evaluate and Display Inline
x+y5=∑k=055k x5−k yk=x5+5⁢x4⁢y+10⁢x3⁢y2+10⁢x2⁢y3+5⁢x⁢y4+y5
Use the Sum and binomial commands to write the unevaluated form of the sum.
Evaluate the sum with the value command.
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