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Algebra
 Factors
  Factoring Special Binomials
  Factoring Special Polynomials
   Binomal Factors of Special Polynomials
   Polynomal Factors of Special Polynomials
   Powers of Binomials
   Powers of Polynomial
 Sources and References

Algebra

Factors

Factoring Special Binomials

Some typical binomial factoring are 𝑎2−𝑏2=(𝑎−𝑏)(𝑎+𝑏) 𝑎3−𝑏3=(𝑎−𝑏)(𝑎2+𝑎𝑏+𝑏2) 𝑎3−𝑏3=(𝑎+𝑏)(𝑎2−𝑎𝑏+𝑏2) In general, 𝑎𝑛−𝑏𝑛=(𝑎−𝑏)(𝑎𝑛−1+𝑎𝑛−2𝑏+⋯+𝑏𝑛−1) Or, if 𝑛 is even 𝑎𝑛−𝑏𝑛=(𝑎+𝑏)(𝑎𝑛−1−𝑎𝑛−2𝑏+⋯−𝑏𝑛−1) And only if 𝑛 is odd. 𝑎𝑛+𝑏𝑛=(𝑎+𝑏)(𝑎𝑛−1−𝑎𝑛−2𝑏+⋯−𝑏𝑛−1)

Factoring Special Polynomials

Binomal Factors of Special Polynomials

Typical polynomials from special factors (𝑥+𝑎)(𝑥+𝑏)=𝑥2+(𝑎+𝑏)𝑥+𝑎𝑏 (𝑥+𝑎)(𝑥+𝑏)(𝑥+𝑐)=𝑥3+(𝑎+𝑏+𝑐)𝑥2+(𝑏𝑐+𝑐𝑎+𝑎𝑏)𝑥+𝑎𝑏𝑐

Polynomal Factors of Special Polynomials

𝑎4+𝑎2𝑏2+𝑏4=(𝑎2+𝑎𝑏+𝑏2)(𝑎2−𝑎𝑏+𝑏2) 𝑎4+𝑏4=(𝑎2+𝑎𝑏√2+𝑏2)(𝑎2−𝑎𝑏√2+𝑏2) 𝑎2+𝑏2−𝑐2+2𝑎𝑏=(𝑎+𝑏)2−𝑐2=(𝑎+𝑏+𝑐)(𝑎+𝑏−𝑐) 𝑎2−𝑏2−𝑐2+2𝑏𝑐=𝑎2−(𝑏−𝑐)2=(𝑎+𝑏+𝑐)(𝑎−𝑏+𝑐) 𝑎3+𝑏3+𝑐3-3𝑎𝑏𝑐=(𝑎+𝑏+𝑐)(𝑎2+𝑏2+𝑐2−𝑏𝑐−𝑐𝑎−𝑎𝑏) 𝑏𝑐2+𝑏2𝑐+𝑐𝑎2+𝑐2𝑎+𝑎𝑏2+𝑎2𝑏+𝑎3+𝑏3+𝑐3=(𝑎+𝑏+𝑐)(𝑎2+𝑏2+𝑐2) 𝑏𝑐2+𝑏2𝑐+𝑐𝑎2+𝑐2𝑎+𝑎𝑏2+𝑎2𝑏+3𝑎𝑏𝑐=(𝑎+𝑏+𝑐)(𝑏𝑐+𝑐𝑎+𝑎𝑏) 𝑏𝑐2+𝑏2𝑐+𝑐𝑎2+𝑐2𝑎+𝑎𝑏2+𝑎2𝑏+2𝑎𝑏𝑐=(𝑏+𝑐)(𝑐+𝑎)(𝑎+𝑏) 𝑏𝑐2+𝑏2𝑐+𝑐𝑎2+𝑐2𝑎+𝑎𝑏2+𝑎2𝑏−2𝑎𝑏𝑐−𝑎3−𝑏3−𝑐3=(𝑏+𝑐−𝑎)(𝑐+𝑎−𝑏)(𝑎+𝑏−𝑐) 𝑏𝑐2−𝑏2𝑐+𝑐𝑎2−𝑐2𝑎+𝑎𝑏2−𝑎2𝑏=(𝑏−𝑐)(𝑐−𝑎)(𝑎−𝑏) 2𝑏2𝑐2+2𝑐2𝑎2+2𝑎2𝑏2−𝑎4−𝑏4−𝑐4=(𝑎+𝑏+𝑐)(𝑏+𝑐−𝑎)(𝑐+𝑎−𝑏)(𝑎+𝑏−𝑐) 𝑥3+2𝑥2𝑦+2𝑥𝑦2+𝑦3=(𝑥+𝑦)(𝑥2+𝑥𝑦+𝑦2) In general, (𝑥+𝑦)𝑛−(𝑥𝑛+𝑦𝑛) is divided by 𝑥2+𝑥𝑦+𝑦2

Powers of Binomials

Some typical polynomals from powers of binomials: (𝑎+𝑏)2=𝑎2+2𝑎𝑏+𝑏2 (𝑎−𝑏)2=𝑎2−2𝑎𝑏+𝑏2 (𝑎+𝑏)3=𝑎3+3𝑎2𝑏+3𝑎𝑏2+𝑏3=𝑎3+𝑏3+3𝑎𝑏(𝑎+𝑏) (𝑎−𝑏)3=𝑎3−3𝑎2𝑏+3𝑎𝑏2−𝑏3=𝑎3−𝑏3−3𝑎𝑏(𝑎+𝑏) Similarly, 𝑥+1𝑥2=𝑥2+2+1𝑥2=𝑥2+1𝑥2+2 𝑥+1𝑥3=𝑥3+3𝑥+1𝑥+1𝑥3=𝑥3+1𝑥3+3𝑥+1𝑥 And Generally, for example 𝑛=7, (𝑎±𝑏)7=𝑎7±7𝑎6𝑏+21𝑎5𝑏2±35𝑎4𝑏3+35𝑎3𝑏4±21𝑎2𝑏5+7𝑎1𝑏6±𝑏7 The next coefficients can be determined by Newton's Rule: Multiply any coefficient by the index ofthe leading quantity, and divide by the number of terms to that plcact to obtain the coefficient of the term next following. i.e. 35=21×5÷3=35×4÷4.

Powers of Polynomial

Some typical polynomials from powers of polynomials: (𝑎+𝑏+𝑐+𝑑)2=𝑎2+2𝑎(𝑏+𝑐+𝑑)+𝑏2+2𝑏(𝑐+𝑑)+𝑐2+2𝑐𝑑+𝑑2  =𝑎2+𝑏2+𝑐2+𝑑2+2𝑎(𝑏+𝑐+𝑑)+2𝑏(𝑐+𝑑)+2𝑐𝑑 (𝑎+𝑏+𝑐)2=𝑎2+𝑏2+𝑐2+2𝑏𝑐+2𝑐𝑎+2𝑎𝑏 (𝑎+𝑏+𝑐)3=𝑎3+𝑏3+𝑐3+3(𝑏2𝑐+𝑏𝑐2+𝑐2𝑎+𝑐𝑎2+𝑎2𝑏+𝑎𝑏2)+6𝑎𝑏 In an algebraical equation, the sign of any letter may be changed throughout, and thus a new formula obtained by keeping an even power of a negative quantity is positive. (𝑎+𝑏−𝑐)2=𝑎2+𝑏2+𝑐2−2𝑏𝑐−2𝑐𝑎+2𝑎𝑏

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive

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ID: 210500028 Last Updated: 5/28/2021 Revision: 0 Ref:

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References

  1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering
  2. Wheatstone, C., 1854, On the Formation of Powers from Arithmetical Progressions
  3. Stroud, K.A., 2001, Engineering Mathematics
  4. Coolidge, J.L., 1949, The Story of The Binomial Theorem
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