Chapter 5: The binomial theorem Notes

Sections

1. 5A Pascal's triangle Mastered

   Expanding (1 + x)^n, building Pascal's triangle with the addition property, its symmetry, and using a row to expand binomials such as (2 + x)^3 and (1 − 2x)^4.

   Open notes
2. 5B Expansions with several variables Mastered

   Expanding (a + b)^n with two variables, binomials such as (2a − 3b)^3 and (x + 1/x)^4, constant terms, and expanding products of binomial powers.

   Open notes
3. 5C The binomial theorem Mastered

   The binomial theorem with sigma notation, why the coefficients are nCk (a counting proof), the identity nCk = (n−1)C(k−1) + (n−1)Ck behind Pascal's triangle, and symmetry.

   Open notes
4. 5D Using the general term Mastered

   The general term T(k+1) = nCk a^(n−k) b^k, finding a particular coefficient, the term independent of x, and the greatest coefficient using the ratio of consecutive terms.

   Open notes
5. 5E Identities in Pascal's triangle Mastered

   Prove identities by substituting values such as x = 1, −1 and 2 into the binomial expansion, and by differentiating it: row sums, alternating sums and weighted sums of coefficients.

   Open notes
6. 5F Further identities in Pascal's triangle Mastered

   Prove identities by equating coefficients of x^r on both sides of an identity such as (1 + x)^n (1 + x)^n = (1 + x)^2n, including the sum of squares of a row, and compare methods of proof.

   Open notes