The two theorems are similar, but refer to different things.
See explanation.
The remainder theorem tells us that for any polynomial ##f(x)##, if you divide it by the binomial ##x-a##, the remainder is equal to the value of ##f(a)##.
The factor theorem tells us that if ##a## is a zero of a polynomial ##f(x)##, then ##(x-a)## is a factor of ##f(x)##, and vice-versa.
For example, let’s consider the polynomial
##f(x) = x^2 – 2x + 1##
Using the remainder theorem
We can plug in ##3## into ##f(x)##.
##f(3) = 3^2 – 2(3) + 1##
##f(3) = 9 – 6 + 1##
##f(3) = 4##
Therefore, by the remainder theorem, the remainder when you divide ##x^2 – 2x + 1## by ##x-3## is ##4##.
You can also apply this in reverse. Divide ##x^2 – 2x + 1## by ##x-3##, and the remainder you get is the value of ##f(3)##.
Using the factor theorem
The quadratic polynomial ##f(x) = x^2 – 2x + 1## equals ##0## when ##x=1##.
This tells us that ##(x-1)## is a factor of ##x^2 – 2x + 1##.
We can also apply the factor theorem in reverse:
We can factor ##x^2 – 2x + 1## into ##(x-1)^2##, therefore ##1## is a zero of ##f(x)##.
Basically, the remainder theorem links the remainder of division by a binomial with the value of a function at a point, while the factor theorem links the factors of a polynomial to its .
Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.
You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.
Read moreEach paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.
Read moreThanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.
Read moreYour email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.
Read moreBy sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.
Read more