Unit 8 of 8
Algebra Used in Integration and Series
- 5 lessons
- Parts 125 to 134
The last unit prepares the algebra of integration and series. It divides polynomials to rewrite a rational function as a polynomial plus a proper fraction, and splits proper fractions into partial fractions as an identity that can be checked.
It then reads indexed terms and shifts an index, works with factorials and binomial coefficients, and reads and rewrites finite sums written with sigma notation.
Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Rewriting Rational Functions
Polynomial Division and Useful Rewrites
How to divide one polynomial by another, write the result as divisor times quotient plus remainder, check it by multiplying back, and keep track of where the fraction form is valid.
Theory part 125Practice part 126
Partial Fractions as an Algebraic Identity
How to split a proper rational expression into simpler fractions with linear and irreducible quadratic denominators, find the unknown numerators, and check the identity on its domain.
Theory part 127Practice part 128
Indices, Factorials, and Sums
Reading Indexed Terms and Shifting the Index
How to read a term formula with an integer index, generate the terms it names, replace n by n + 1 everywhere at once, and simplify the ratio of consecutive terms.
Theory part 129Practice part 130
Factorials and Binomial Coefficients
What n! and the binomial coefficient mean for nonnegative integers, how to cancel factorials in a ratio safely, and why choosing positions in a product gives the coefficients of a binomial expansion.
Theory part 131Practice part 132
Reading and Rewriting Finite Sums
How to read sigma notation as a finite row of terms, count terms from the bounds, rename or shift the index without changing the sum, and spot a telescoping sum.
Theory part 133Practice part 134