Pascal triangle and binomial theorem

Pascal triangle and binomial theorem

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  PASCAL’S TRIANGLE AND THEBINOMIAL THEOREM PreCalculus
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  PASCAL’S TRIANGLE •Can yourecall the formula for the square of a binomial? (x+ y)2 = How about the cube of a binomial? (x + y)3 =
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  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
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  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
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  PASCAL’S TRIANGLE If wecontinue 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.
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  Let us lookat 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
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  Let us lookat 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.
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  BINOMIAL THEOREM If nis 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 !
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  EXAMPLE 1 •Expand (x+y)5using the binomial theorem.
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  EXAMPLE 2 •Expand (x-y)5.
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  EXAMPLE 3 •Expand (2x-y)6using the binomial theorem.
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  FINDING THE COEFFICIENTUSING 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
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  EXAMPLE 4 •What isthe fourth term in the expansion of (a+b)5?
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  EXAMPLE 5 •Determine thefifth term in the expansion of (x+2y)6 ?
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  Reference: • Aoanan, GraceO. 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
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