Example
Multiply 3x^{5} + 4x^{3} + 2x - 1 and x^{4} + 2x^{2} + 4.
The product is given by
3x^{5} . (x^{4} + 2x^{2} + 4) + 4x^{3}. (x^{4} + 2x^{2} + 4) + 2x .
(x^{4} + 2x^{2} + 4) - 1 . (x^{4} + 2x^{2} + 4)
= 3x^{5 }. x^{4} + 3x^{5} . 2x^{2} + 3x^{5} . 4 + 4x^{3} . x^{4} + 4x^{3} .
2x^{2 }+ 4x^{3 }. 4 + 2x . x^{4} + 2x . 2x^{2} + 2x . 4 - x^{4 }- 2x^{2} - 4
To simplify the above we employ a rule which we will learn in laws of indices. It states that x^{m} . x^{n} = x^{m+n}
= 3x^{9} + 6x^{7} + 12x^{5} + 4x^{7} + 8x^{5} + 16x^{3} + 2x^{5} + 4x^{3} + 8x - x^{4} - 2x^{2} - 4
Now we collect like terms and simplify them. We obtain 3x^{9} + 10x^{7} + 22x^{5} - x^{4} + 20x^{3} - 2x^{2} + 8x - 4.