Appendix
A-4: Algebraic Expressions and Operations
Example A-4.6
From first principles, expand x+y5 by the Binomial theorem.
Solution
Mathematical Solution
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.
x+y5
=∑k=055k x5−k yk
=50x5y0+51x4y1+52x3y2+53x2y3+54x1y4+55x0y5
=5!0! 5!x5+5!1! 4!x4y+5!2! 3!x3y2+5!3! 2!x2y3+5!4! 1!xy4+5!5! 0!y5
=x5+5x4y+10x3y2+10x2y3+5xy4+y5
Interactive Solution
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+5x4y+10x3y2+10x2y3+5xy4+y5
Coded Solution
Use the Sum and binomial commands to write the unevaluated form of the sum.
q≔Sumbinomial5,k⋅x5−k⋅yk,k=0..5
Evaluate the sum with the value command.
valueq
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