Pascal triangle and binomial theorem
Download as PPTX, PDF7 likes2,044 views
Pascal's triangle is a triangular array of the binomial coefficients that arises from the binomial formulas. It was studied extensively by the French mathematician Blaise Pascal in the 17th century. The binomial theorem states that the expansion of (a + b)^n can be written as the sum of terms involving the binomial coefficients, with the coefficient of each term found using the appropriate entry in Pascal's triangle. Examples are provided of using the binomial theorem to expand expressions like (x + y)^5 and determining coefficients of specific terms in the expansions.
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
Editor's Notes
- #8: These observations on binomial expansion will be generalized binomial theorem.