7. Binomial Theorem

4 topics51 min

Chapter 7 of NCERT Mathematics for Class 11: Binomial Theorem. Two sections — Where the coefficients come from and The theorem, and what it hands you. Four videos, 51 min in all.

Worked answers to the 20 exercise questions in this chapter: Exercise 7.1 · Miscellaneous Exercise

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Part 1

Where the coefficients come from

13 minReading the pattern out of the first few expansionsThe row of coefficients for the fourth power is the third power's row, copied, shifted one place along, and added to itself. Even the 1 at both ends comes from that.13 min · 12 parts12 minRewriting the triangle with selection counts, so any row is reachable directlyTo write the coefficients for the twelfth power, the triangle first demands the eleven rows above it — ninety-one numbers for the thirteen wanted. One rewrite needs none.12 min · 12 parts

Part 2

The theorem, and what it hands you

12 minProving the expansion for every positive power by inductionA rule built to agree with every coefficient row checked so far does agree, on all of them — and first breaks at the sixth power. Checking rows is not proving a pattern.12 min · 13 parts14 minSubstituting particular values, and the coefficient identities that drop outSet both terms in an expansion to one, and its row of coefficients sums to a power of two. Negate the second term first, and the same row sums to nothing at all.14 min · 13 parts

What you will be able to do

  • Write out the expansions of a two-term bracket for indices 0 to 4 and say what the page prints beside the zeroth one, and why that condition is there
  • State why building the row for a large index from the triangle alone is impractical, using the chapter's own twelfth-power case
  • State the expansion §7.2.1 asserts, and name the two things it ranges over
  • Derive the signed expansion by substituting the negative of the second term, and say which positions change sign and why

What this chapter assumes you already know

  • Squaring and cubing a two-term bracket, from earlier algebra
  • Multiplying a polynomial by a two-term bracket, and collecting like terms
  • Laws of exponents for a product and for a power of a power, since a bracket such as 2x has to be raised whole
  • That an index counts repeated multiplication, and that raising anything to the power 1 leaves it alone
  • Multiplying out a product of several brackets without collecting like terms
  • Multiplying a many-term expression by a two-term bracket, and collecting like terms
  • What it means to prove a statement for every member of an unbounded list, as against checking a few of them
  • Laws of exponents applied to a negative quantity, so that even and odd powers can be told apart

It builds directly on Every selection was counted once per arrangement of itself, Choosing what to leave out, and the rule that builds each count from two smaller ones and A shorthand for descending products, and why the empty product is set to one.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "The coefficients are just a sequence to memorise."
  • "Row 5 is printed in Fig 7.1 or Fig 7.2."
  • "Raising 2x to the fourth gives 2x⁴."
  • "The condition beside the zeroth power is a typographical leftover."
  • "1 sits at the ends because the pattern looks nicer that way."
  • "The rows are symmetric by coincidence."
  • "The triangle solves the problem."
  • "The rewriting is cosmetic — the same numbers in fancier clothes."