Algebra

Highest Common Factor

Rule: To find the highest common factor of two expressions: Divide the one which is of the highest dimension by the other, rejecting first any factor of either expression which is not also a factor of the other. Operate in the same manner upon the remainder and the divisor, and continue the process until there is no remainder. The last divisor will be the highest common factor required.

Example

to find the H.C.F. of 3𝑥⁵−10𝑥³+15𝑥+8 and 𝑥⁵−2𝑥⁴−6𝑥³−4𝑥²+13𝑥+6.

𝑥 |  5  4  3  2  1  0 |  5 4  3  2  1  0 | 
- | ----------------- | ---------------- | --
  |  1− 2− 6+ 4+13+ 6 |  3+0−10+ 0+15+ 8 |  3
  |  3× | −3+6−18−12−39−18 | 
  | ----------------- | ---------------- | 
𝑥 |  3− 6−18+12+39+18 |  2)6+ 8−12−24−10 | 
  | −3− 4+ 6+12+ 5 |    3+ 4− 6−12− 5 | 
  | ----------------- |   | 
  |  2)−10−12+24+44+18 |   | 
  |    − 5− 6+12+22+ 9 |   | 
  |      3 |   | 
  | ----------------- |   | 
5 |    −15−18+36+66+27 |   | 
  |    +15+20−30−60−25 |   | 
  | ----------------- | ---------------- | 
  |      2) 2+ 6+ 6+ 2 |    3+ 4− 6−12− 5 | 3𝑥
  |         1+ 3+ 3+ 1 |   −3− 9− 9− 3 | 
  |   | ---------------- | 
  |   |     − 5−15−15− 5 | −5
  |   |     + 5+15+15+ 5 | 
  |   | ---------------- | 

Evolution

Otherwise: To form the H.C.F. of two or more algebraical expressions: Separate the expressions into their simplest factors. The H.C.F. will be the product of the factors common to all the expressions, taken in the lowest powers that occur.

Lowest Common Multiple

The L.C.M. of two quantities is equal to their product divided by the H.C.F. Otherwsise.: To form the L.C.M. of two or more algebraical expressions: Separate them into their simplest factors. The L.C.M. will be the product of all the factors that occur, taken in the highest powers that occur.

Example

The H.C.F. of 𝑎²(𝑏−𝑥)⁵𝑐⁷𝑑 and 𝑎³(𝑏−𝑥)²𝑐⁴𝑒 is 𝑎²(𝑏−𝑥)²𝑐⁴; the L.C.M. is 𝑎³(𝑏−𝑥)⁵𝑐⁷𝑑𝑒

Evolution

Square Root

To extract the Square Root of 𝑎²−3𝑎√𝑎2−3√𝑎2+41𝑎16+1 16𝑎²−24𝑎³²+41𝑎−24𝑎¹²+1616 Detaching the coefficients, the work is as follows:

𝑎 |   2  32  1 12 0 1 12 0
        ----- | --------------
          |  16−24+41−24+16 ( 4-3+4
        4 | −16
          | --------------
        2×4 |    −24+41
        8-3 |     24− 9
          | --------------
        8-2×3 |       32−24+16
        8-6+4 |      −32+24−16

⇒root: 4𝑎−3𝑎^1/2+44=𝑎−34√𝑎+1

Cube Root

To extract the Cube Root of 8𝑥⁶−36𝑥⁵√𝑦+66𝑥⁴𝑦−63𝑥³𝑦√𝑦+33𝑥²𝑦²−9𝑥𝑦²√𝑦+𝑦³ The terms here contain the successive powers of 𝑥 and √𝑦; therefore, detaching the coefficients, the work will be as follow:

4 3 |2 1 0|4 3 2 1 0| 6  5  4  3  2 1 0|2 1 0
        ---|-----|---------|------------------
         | | | 8−36+66−63+33−9+1(2−3+1
        22|3×2|3×22|−8⇒2
         | | |------------------
         | | |  −36+66−63+33−9+1
        3×22|3×2−3|3×22−3×2(3)+(−3)2|  +36⇒−3
         | | |------------------
         | | |     +66−63+33−9+1
         | |0−18+9|     −54+27
         | | |------------------
         | | |     +12−36+33−9+1
        3×22|3×2(1)−3×3(1)+12|0−2x3×2(3)+3×(−3)2|     −12⇒1
         | | |------------------
         | | |        −36+33−9+1
         | |0−36+27|        +36−27
         |6−9+1| |           −06+9−1

⇒root:2𝑥²−3𝑥√𝑦+𝑦 The foregoing process is but a slight variation of Horner's rule for solving an equation of any degree

Sources and References

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