Synthetic Multiplication

Synthetic multiplication is a method of multiplying polynomials by considering only the coefficients. The coefficients can be considered as forming ordinary numbers and multiplied in the normal way, with the result of the multiplication being translated back into a polynomial.

Consider the product

Expanding the brackets gives, after some simplification,

Now write the polynomialas 213 and the polynomialas 122,


2, 5, 9, 8 and 6 are the coefficients ofand 6 is the constant term respective,ly so we can write down the result of multiplying the two polynomials as

Synthetic multiplication is limited and cannot handle carried digits so that we would not be abke to multiplyby

In fact

Using synthetic multiplication, writingas 47 andas 36, we obtain 47*36=1692, which would give the polynomialwhich is obviously wrong.