• Skip to main content
  • Skip to secondary menu
  • Skip to primary sidebar
  • Skip to footer
  • ICSE Solutions
    • ICSE Solutions for Class 10
    • ICSE Solutions for Class 9
    • ICSE Solutions for Class 8
    • ICSE Solutions for Class 7
    • ICSE Solutions for Class 6
  • Selina Solutions
  • ML Aggarwal Solutions
  • ISC & ICSE Papers
    • ICSE Previous Year Question Papers Class 10
    • ISC Previous Year Question Papers
    • ICSE Specimen Paper 2021-2022 Class 10 Solved
    • ICSE Specimen Papers 2020 for Class 9
    • ISC Specimen Papers 2020 for Class 12
    • ISC Specimen Papers 2020 for Class 11
    • ICSE Time Table 2020 Class 10
    • ISC Time Table 2020 Class 12
  • Maths
    • Merit Batch

A Plus Topper

Improve your Grades

  • CBSE Sample Papers
  • HSSLive
    • HSSLive Plus Two
    • HSSLive Plus One
    • Kerala SSLC
  • Exams
  • NCERT Solutions for Class 10 Maths
  • NIOS
  • Chemistry
  • Physics
  • ICSE Books

The Binomial Theorem

December 9, 2020 by sastry

The Binomial Theorem

You are faced with the problem of expanding \((x+y)^10\) . What to do??? Do you really have to multiply this expression times itself 10 times?? That could take forever.

Let’s investigate:

When binomial expressions are expanded, is there any type of pattern developing which might help us expand more quickly? Consider the following expansions:
The Binomial Theorem 1
What observations can we make in general about the expansion of \((a+b)^n\)

1. The expansion is a series (an adding of terms).
2. The number of terms in each expansion is one more than n. (terms = n + 1)
3. The power of a starts with an and decreases by one in each successive term ending with a0. The power of b starts with b0 and increases by one in each successive term ending with bn.
4. The power of b is always one less than the “number” of the term. The power of a is always n minus the power of b.
5. The sum of the exponents in each term adds up to n.
6. The coefficients of the first and last terms are each one.
7. The coefficients of the middle terms form an interesting (but perhaps not easily recognized) pattern where each coefficient can be determined from the previous term. The coefficient is the product of the previous term’s coefficient and a’s index, divided by the number of that previous term.

The Binomial Theorem 2

8. Another famous pattern is also developing regarding the coefficients. If the coefficients are “pulled off” of the terms and arranged, they form a triangle known as Pascal’s triangle. (The use of Pascal’s triangle to determine coefficients can become tedious when the expansion is to a large power.)

The Binomial Theorem 3

(The two outside edges of the triangle are comprised of ones. The other terms are each the sum of the two terms immediately above them in the triangle.)

By pulling these observations together with some mathematical syntax, a theorem is formed relating to the expansion of binomial terms:

The Binomial Theorem 4

The Binomial Theorem 5

Examples using the Binomial Theorem:
The Binomial Theorem 6

The Binomial Theorem 7

Filed Under: Mathematics Tagged With: The Binomial Theorem

Primary Sidebar

  • MCQ Questions
  • RS Aggarwal Solutions
  • RS Aggarwal Solutions Class 10
  • RS Aggarwal Solutions Class 9
  • RS Aggarwal Solutions Class 8
  • RS Aggarwal Solutions Class 7
  • RS Aggarwal Solutions Class 6
  • ICSE Solutions
  • Selina ICSE Solutions
  • Concise Mathematics Class 10 ICSE Solutions
  • Concise Physics Class 10 ICSE Solutions
  • Concise Chemistry Class 10 ICSE Solutions
  • Concise Biology Class 10 ICSE Solutions
  • Concise Mathematics Class 9 ICSE Solutions
  • Concise Physics Class 9 ICSE Solutions
  • Concise Chemistry Class 9 ICSE Solutions
  • Concise Biology Class 9 ICSE Solutions
  • ML Aggarwal Solutions
  • ML Aggarwal Class 10 Solutions
  • ML Aggarwal Class 9 Solutions
  • ML Aggarwal Class 8 Solutions
  • ML Aggarwal Class 7 Solutions
  • ML Aggarwal Class 6 Solutions
  • HSSLive Plus One
  • HSSLive Plus Two
  • Kerala SSLC

Recent Posts

  • Air Pollution Essay for Students and Kids in English
  • 10 Lines on Satya Nadella for Students and Children in English
  • Essay on Wonders of Science | Wonders of Science for Students and Children in English
  • My Childhood Memories Essay | Essay on My Childhood Memories for Students and Children in English
  • Essay On Sports And Games | Sports And Games Essay for Students and Children in English
  • Clean India Slogans | Unique and Catchy Clean India Slogans in English
  • Animals that Start with Z | Listed with Pictures, List of Animals Starting with Z & Interesting Facts
  • Positive Words That Start With S | List of Positive Words Starting With S Meaning, Examples,Pictures and Facts
  • Positive Words that Start with B | List of 60 Positive Words Starting with B Pictures and Facts
  • Humanity Essay | Essay on Humanity for Students and Children in English
  • Teenage Pregnancy Essay | Essay on Teenage Pregnancy for Students and Children in English

Footer

  • RS Aggarwal Solutions
  • RS Aggarwal Solutions Class 10
  • RS Aggarwal Solutions Class 9
  • RS Aggarwal Solutions Class 8
  • RS Aggarwal Solutions Class 7
  • RS Aggarwal Solutions Class 6
  • Picture Dictionary
  • English Speech
  • ICSE Solutions
  • Selina ICSE Solutions
  • ML Aggarwal Solutions
  • HSSLive Plus One
  • HSSLive Plus Two
  • Kerala SSLC
  • Distance Education
DisclaimerPrivacy Policy
Area Volume Calculator