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The Binomial Theorem: Expand and Evaluate Binomial Expressions

Learn how to expand and evaluate binomial expressions using the binomial theorem. Discover the pattern in Pascal's Triangle and understand the general term using factorials.

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The Binomial Theorem: Expand and Evaluate Binomial Expressions

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  1. 15 Minutes – 4 questions 2.(i) -15,0,3,0,-3,0,15 (ii) x (x+2) (x-2) (iii) -2, 0, 2 4. (i) f(3)=0 (ii) x=-2,3 or 4 C1/C2 Ex 3c page 95 Nos 2, 4 C1/C2 Ex 3C page 95 Nos 12, 13 13. -12 12. (ii) a) -6 b) -2 c) -2x-4 ML5 MH

  2. Lesson 5-The Binomial Theorem Objectives : - To be able to expand (x+y)n quickly - The Binomial Theorem is useful in statistics - To learn and use the nCr - To be able to expand (x+y)-n quickly - To be able to expand (x+y)m/n quickly ML5 MH

  3. The Binomial Theorem 2 Terms like (1+2x) are referred to as being Binomial KEYWORDS Binomial Coefficient Expansion The binomial theorem is used to raise a binomial (a + b)99 to large powers.

  4. The Binomial Theorem (Binomial expansion) coefficient (a + b)2 ≡ (a + b)(a + b) ≡1a2 + 2ab + 1b2 (a + b)1 ≡ 1a +1b (a + b)3 ≡ (a + b)(a + b)(a + b) ≡1a3 + 3a2b +3ab2 +1b3

  5. The Binomial Theorem (Binomial expansion) (a + b)4 ≡ (a + b)(a + b)(a + b)(a +b) =1a4 + 4a3b +6a2b2 +4ab3+1b4 Take out the coefficients of each expansion. 2 1 1 11 11 2 1 3 3 1 1 1 3 1 3 1 4 6 4 1 1 4 4 6 1

  6. + + + + The Binomial Theorem (Binomial expansion) Can you guess the expansion of (a + b)5 without expanding the factors ? (a + b)5 =1a5 + 5a4b +10a3b2 +10a2b3+5ab4+1b4

  7. Pascal's Triangle

  8. Expand (1+x)6Using Pascal’s ∆ • (1+x)5 ≡ 1 + 5x + 10x2 + 10x3 +5x4 + 1x5 • (1+x) 6 ≡ 1 + 6x + 15x2 + 20x3 + 15x4 + 6x5 + 1x6 • (1-x) 6 ≡ 1 + 6(-x) + 15(-x)2 +20(-x)3 + 15(-x)4 + 6(-x)5 +1(-x)6 • ≡ 1 - 6x + 15x2 - 20x3 + 15x4 - 6x5 +1x6 • Odd exponentsbecome negative as we substitute x for -x what about (1-x)6 ML5 MH

  9. The triangle first appeared in ancient Chinese Math Texts. The Chinese had known about the binomial theorem for a long time. Because Pascal was for so long (until 1935) the first known discoverer of the triangle in the Western World, and because of his development and application of the triangle’s properties, the array became known as Pascal’s Triangle. Brief History of the Triangle(1635 probability) Daria Islam, Columbia University Graduate School of Education

  10. Let’s Construct the Triangle Together

  11. (a+b)5 = a5+5a4b+10a3b2+10a2b3+5ab4+b5 Points to be noticed : • Coefficients are arranged in a Pascal triangle. • Summation of the indicesof each term is equal to the power (order) of the expansion. • The first term of the expansion is arranged in descending order after the expansion. • The second termof the expansion is arranged in ascending orderorder after the expansion. • Number of termsin the expansion is equal to the power of the expansion plus one.

  12. This would still be inconvenient and awkward We can make things easier if we can recognise the pattern and write the coefficients in a General form

  13. The General TermIntroducing the FACTORIAL ! Factorial - The productof the first n positive integers i.e.n！= n(n-1)(n-2)(n-3)….3×2×1 0！is defined to be 1. i.e. 0！= 1

  14. The General Term • (1+x) 6 ≡ 1 + 6x + 15x2 + 20x3 + 15x4 + 6x5 + 1x6 Consider term 3 and term 5 These can be written using Factorial as : n=6 r=2 (3rd term) n=6 r=4(5th term) ML5 MH

  15. nCr = 5C2 = 5C2 = 5C2 =

  16. The general Term- using ! (1+x) 6 ≡ 1 + 6x + 15x2 + 20x3 + 15x4 + 6x5 + 1x6 ≡ 6C0 + 6C1x + 6C2 x2 + 6C3 x3 + 6C4x4 + 6C5 x5 + 6C6x6 ML5 MH

  17. “N” “C” “R” • N-C-R is written in the following way ML5 MH

  18. General Term This formular needs to be understood This information is on your formula sheet SIGMA ML5 MH

  19. Examples • a) Expand • b) Hence find • Solution , here n=3; x=(5x) so ML5 MH

  20. Examples • b) Expand • Same as previous BUT replace x with (-x) ML5 MH

  21. Exercises 3F page 115 C1/C2 ML5 MH

  22. Another Example Hint, let x=0.01 ML5 MH

  23. Try these • Expand n=4 ; x=-1/2x • Solution ML5 MH

  24. Negative Indices • This formula is still valid for negative exponent • Example : • Write down the expansion of {use n=-2; x=-x in 1} 1. ML5 MH

  25. Example – negative fraction • Expand the following Let n=-1/2 x=2t ML5 MH

  26. Expand for the first 4 terms • First rewrite as • Expand the binomial, then multiply by Here n=-2; x=-2/3 ML5 MH

  27. n=-2 ; x=-2/3 Evaluate terms Multiply by 1/9 Final solution Exercise 7A C3/C4 Page 164 ML5 MH

  28. Summary • The binomial theorem gives a quick way of expanding high power terms like (1-x)35 • For positive powers, fractions and negative powers use this formula Substitute for n and x in (1+x)n ; (1-x)n ; (1-x)-n • For positive N can use nCron your calculator to • find expansion quickly ML5 MH

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