Algebra

Multiplication and Division

By the method of detacched coefficients

Multiplication

By the method of detacched coefficients Example:(𝑎⁴−3𝑎²𝑏²+2𝑎𝑏³+𝑏⁴)×(𝑎³−2𝑎𝑏²−2𝑏³) 1+0−3+2+1 1+0−2−2 ------------------------------- 1+0−3+2+1     −2−0+6−4−2       −2−0+6−4−2 ------------------------------- 1+0−5+0+7+2−6−2 ⇒𝑎⁷−5𝑎⁵𝑏²+7𝑎³𝑏⁴+2𝑎²𝑏⁵−6𝑎𝑏⁶−2𝑏⁷

Division

By the method of detacched coefficients

Example:(𝑥⁷−5𝑥⁵+7𝑥³+2𝑥²−6𝑥−2)÷(𝑥⁴−3𝑥²+2𝑥+1) 1+0−3+2+1)  1+0−5+0+7+2−6−2(1+0−2−2  −1−0+3−2−1  -------------------------------      0−2−2+6+2−6        +2+0−6+4+2  -------------------------------          −2+0+6−4−2          +2+0−6+4+2 ⇒𝑥³−2𝑥−2

By the method of synthetic division

Example:(𝑥⁷−5𝑥⁵+7𝑥³+2𝑥²−6𝑥−2)÷(𝑥⁴−3𝑥²+2𝑥+1)

 | 𝑥| 7 6 5 4 3 2 1 0
𝑥| | 1+0−5+0+7+2−6−2
3|−0|   0+0+0+0
 | | 1+0−5+0+7+2−6−2
2|+3|    +3+0−6−6
 | | 1+0−2+0+1−4−6−2
1|−2|      −2+0+4+4
 | | 1+0−2−2+1+0−2−2
0|−1|        −1+0+2+2
 | |-------------------------------
 | | 1+0−2−2+0+0+0+0
4|−1|−1+0+2+2
 | |-------------------------------
 | | 0+0+0+0+0+0+0+0
 | |-------------------------------
 | |1+0−2−2
 | 𝑥|3 2 1 0

⇒𝑥³−2𝑥−2 Note that, in all operations with detached coefficients, the result must be written out in successive powers of the quantity which stood in its successive powers in the original expression.

Sources and References

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