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Binomial Forms

Binomial Forms. Expansion of Binomial Expressions. Polynomial Functions. Special Forms Difference of Squares (Conjugate Binomials) a 2 – b 2 = ( a – b)( a + b) Difference of n th Powers a n – b n = ( a – b)( a n–1 + a n–2 b + a n–3 b 2 + ... + a b n–1 + b n–1 ).

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Binomial Forms

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  1. Binomial Forms Expansion of Binomial Expressions

  2. Polynomial Functions • Special Forms • Difference of Squares (Conjugate Binomials) a2 – b2=(a – b)(a+ b) • Difference of nth Powers an – bn = (a – b)(an–1+an–2b +an–3b2+ ... +abn–1+ bn–1) Binomial Forms

  3. n(n–1)...(2) abn–1 + bn + (n – 1)! n(n – 1) ... an–2b2 + 2 Polynomial Functions • Special Forms • Perfect-Square Trinomials • (a+ b)2 = a2+ 2ab + b2 • (a – b)2 = a2 – 2ab + b2 • Binomial Expansion (a+ b)n = an+ nan–1b + Binomial Forms

  4. Binomial Theorem • Expansion of (a+b)n • Examples: • (a+b)2 = a2+ 2ab+b2 • (a+b)3 = a3+ 3a2b+ 3ab2+b3 • (a+b)4 = a4+ 4a3b+ 6a2b2 + 4ab3+b4 Binomial Forms

  5. 5 4 3 • • a5+ a4b + a3b2 + a2b3 5 54 3 2 1 • • • 2 1 • 1 • 5! 5 4 3 2 • • • 5! 4 3 2 1 • • • ab4 b5 + + • • • n!= n(n – 1)(n – 2)(n – 3) 3 2 1 • • Binomial Theorem • Expansion of (a+b)n • Examples: (a+b)5 = = a5+ 5a4b+ 10a3b2+ 10a2b3+ 5ab4+b5 n factorial,n!, is defined as Note: and 0! ≡ 1 by definition Binomial Forms

  6. an–2b2 (a+b)n= an+nan–1b … bn + + n ∑ n! an–kbk = n(n – 1) • • • + (n–k)! k! 2 k=0 n ∑ ak = a0+a1+a2++ an k=0 Binomial Theorem • Expansion of (a+b)n • The General Case: • A Note: Binomial Forms

  7. Binomial Theorem • Expansion of (a+b)n • Pascal’s Triangle • Coefficients for (a+b)n Degree 1 n = 0 n = 1 1 1 1 2 1 n = 2 1 1 3 3 1 n = 3 1 1 4 6 4 1 n = 4 Fibonacci Sequence 2 1 5 10 10 51 n = 5 3 . . . . . . . . 5 8 Binomial Forms

  8. Binomial Theorem 1 1 • Expansion of (a+b)n • Pascal’s Triangle • Examples: • (a+b)0 = 1 1 1 1 1 2 1 1 2 1 1 3 3 1 1 3 3 1 1 4 6 4 1 1 4 6 4 1 1 1 5 10 10 5 1 1 5 10 10 5 1 1•a+ 1•b • (a+b)1 = . . . . . . . . 1•a2+ 2ab+ 1•b2 • (a+b)2 = 1•a3+ 3a2b+ 3ab2+ 1•b3 • (a+b)3 = 1•a4+ 4a3b+ 6a2b2+ 4ab3+ 1•b4 • (a+b)4 = • (a+b)5 = 1•a5+ 5a4b + 10a3b2+ 10a2b3+ 5ab4 + 1•b5 Binomial Forms

  9. Thinkabout it ! Binomial Forms

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