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# Content

`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

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

1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science &amp; 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  Home 5

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