Remainder Theorem
and Factor Theorem

Or: how to avoid Polynomial Long Division when finding factors

Do you remember doing division in Arithmetic?

"7 divided by 2 equals 3 with a remainder of 1"

Each part of the division has names:

Which can be rewritten as a sum like this:

Polynomials

Well, we can also divide polynomials.

f(x) ÷ g(x) = q(x) with a remainder of r(x)

But it is better to write it as a sum like this:

Like in this example using Polynomial Long Division:

Example: 2x²-5x-1 divided by x-3

• f(x) is 2x²-5x-1
• g(x) is x-3

After dividing we get the answer 2x+1, but there is a remainder of 2.

• q(x) is 2x+1
• r(x) is 2

In the style f(x) = g(x)·q(x) + r(x) we can write:

2x²-5x-1 = (x-3)(2x+1) + 2

https://www.mathsisfun.com/algebra/polynomials-remainder-factor.html