Pascal triangle and binomial theorem
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Binomial expansion using Pascal's triangle and binomial theorem
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Pascal triangle and binomial theorem
- 1. PASCAL’S TRIANGLE AND THE BINOMIAL THEOREM PreCalculus
- 2. PASCAL’S TRIANGLE •Can you recall the formula for the square of a binomial? (x+ y)2 = How about the cube of a binomial? (x + y)3 =
- 3. PASCAL’S TRIANGLE •In his “Treatise on the Arithmetical Triangle” of 1653, French mathematician Blaise Pascal (1623- 1662) wrote extensively about a triangular array of numbers for binomial coefficients, now called
- 4. PASCAL’S TRIANGLE For (x + y)o 1 For (x + y)1 1 1 For (x + y)2 1 2 1 For (x + y)3 1 3 3 1 For (x + y)4 1 4 6 4 1
- 5. PASCAL’S TRIANGLE If we continue Pascal’s triangle above, we have For (x + y)5 1 5 10 10 5 1 Thus, we have formulas for (x+y)4 and (x+y)5.
- 6. Let us look at each term of the expansion (x+y)4=x4+4x3y+6x2y2+4xy3+y4 , as shown below. (x+y)4 exponent of x exponent of y sum of the exponents x4 4 0 4 4x3y 3 1 4 6x2y2 2 2 4 3
- 7. Let us look at each term of the expansion (x+y)4=x4+4x3y+6x2y2+4xy3+y4 , as shown below. (x+y)4 exponent of x exponent of y sum of the exponents x4 4 0 4 4x3y 3 1 4 6x2y2 2 2 4 3 These observations on binomial expansion will be generalized binomial theorem.
- 8. BINOMIAL THEOREM If n is a positive integer, then 𝒂 + 𝒃 𝒏 = 𝒏 𝟎 𝒂 𝒏 + 𝒏 𝟏 𝒂 𝒏−𝟏 𝒃 + 𝒏 𝟐 𝒂 𝒏−𝟐 𝒃 𝟐 + 𝒏 𝟑 𝒂 𝒏−𝟑 𝒃 𝟑 + ⋯ + 𝒏 𝒏 − 𝟏 𝒂𝒃 𝒏−𝟏 + 𝒏 𝒏 𝒃 𝒏 𝒐𝒓 𝒂 + 𝒃 𝒏 = 𝒌=𝒐 𝒏 𝒏 𝒌 𝒂 𝒏−𝒌 𝒃 𝒌 , Where 𝒏 𝒌 = 𝒏! 𝒌! 𝒏−𝒌 !
- 9. EXAMPLE 1 •Expand (x+y)5 using the binomial theorem.
- 11. EXAMPLE 3 •Expand (2x-y)6 using the binomial theorem.
- 12. FINDING THE COEFFICIENT USING BINOMIAL THEOREM •If t is the exponent of b, then (t +1)st term contains a(n-t)bt and its coefficient is. 𝑛 𝑛 − 1 𝑛 − 2 ∙∙∙ 𝑛 − (𝑡 − 1) 𝑡 𝑡 − 1 (𝑡 − 2) ∙∙∙ 1
- 13. EXAMPLE 4 •What is the fourth term in the expansion of (a+b)5?
- 14. EXAMPLE 5 •Determine the fifth term in the expansion of (x+2y)6 ?
- 15. Reference: • Aoanan, Grace O. et. al. (2018) General Mathematics for Senior HS, C&E Publishing, Inc. • Garces, Ian June L. et. al. (2016) Precalculus “Teaching Guide for Senior High School, Commission on Higher
Hinweis der Redaktion
- These observations on binomial expansion will be generalized binomial theorem.