To understand multiplication of polynomials by long multiplication, first consider the sample below:
(2x + 3)(4x - 5) = 8x
2
+ 2x – 15 as was shown in the explanation of multiplication of polynomials using the distributive law.
Now, consider the following process by which the polynomials to be multiplied are written on different lines like in multiplication of numbers. Note that the polynomials should be in standard format with descending powers. After writing the polynomials on separate lines, the highest power term polynomial of the on the lower line is multiplied by the highest power term of the other polynomial, and the result is recorded on the third line. The highest power term of the polynomial on the lower line of the second-highest power term of the other polynomial, and this result is recorded on the third line to the right of the number already recorded. This process is repeated until all terms of the other polynomial have been used. Then the second term of the polynomial on the second line is multiplied with recordings as above and so on until all of the products are on appropriate lines.
A very important point is to be certain that the product terms obtained are recorded with the same powers in vertical columns.
As an example from the process of multiplication using the distributive law, this now becomes:
2x + 3
* 4x - 5
8x
2
+ 12x
- 10x - 15
8x
2
+ 2x - 15
Similarly, for (1x + 2)(3x
2
+ 4x + 5) = 9x
2
+ 17x +10. Notice that for this example, it is practical to put the shorter polynomial on the second line.
So, we have
3x
2
+ 4x + 5
* x + 2
3x
3
+ 4x
2
+ 5x
6x
2
+ 8x + 10
3x
3
+ 10x
2
+ 13x + 10
For multiplication of short polynomials, using the distributive law as in FOIL will normally be the method of choice, but for polynomials of degree 3 or higher, long multiplication will normally reduce the opportunity for errors.