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, 𝑎^𝑛−𝑏^𝑛=(𝑎−𝑏)(𝑎^𝑛−1+𝑎^𝑛−2𝑏+⋯+𝑏^𝑛−1) Or, if 𝑛 is even 𝑎^𝑛−𝑏^𝑛=(𝑎+𝑏)(𝑎^𝑛−1−𝑎^𝑛−2𝑏+⋯−𝑏^𝑛−1) And only if 𝑛 is odd. 𝑎^𝑛+𝑏^𝑛=(𝑎+𝑏)(𝑎^𝑛−1−𝑎^𝑛−2𝑏+⋯−𝑏^𝑛−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−(𝑥^𝑛+𝑦^𝑛) 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