The Remainder Theorem and The Factor Theorem

[Stratosphere]

Homework Statement

I understand How to do The remainder Theorem and The factor Theorem but I don't understand what they mean or what they are doing. I don't think I will be able to apply them without knowing what they mean. Can someone explain them to me?

Homework Equations

The Attempt at a Solution

 

[rock.freak667]

The theorem is basically this, if f(x) is any polynomial and f(x) is divided by x-a, then the remainder is f(a) [remainder theorem]. If f(a)=0 then x-a is factor of f(x) [factor theorem]

so basically when you divide f(x) by x-a, you will get a polynomial p(x) and a remainder R. So you can rewrite f(x) as such

f(x)=p(x)*(x-a)+R

clearly you can see why, the remainder is f(a).
You can also see that when f(a)=0, the remainder R=0 as well. This can only happen when you divide by a factor.

I hope this explained it better for you.

 

[Architeuthis Dux]

The Remainder Theorem and The Factor Theorem are two important concepts in algebra that help us understand and solve polynomial equations. The Remainder Theorem states that when a polynomial f(x) is divided by (x-a), the remainder is equal to f(a). This means that if we have a polynomial function and we divide it by (x-a), the value we get when we substitute a into the function will be the remainder. This can be useful in determining the remainder of a polynomial division and in solving problems involving finding factors of a polynomial.

The Factor Theorem, on the other hand, states that if the polynomial f(x) has a factor (x-a), then f(a) = 0. This means that if we have a polynomial function and we plug in the value of a into the function and get 0, then (x-a) is a factor of the polynomial. This can be useful in finding factors of a polynomial and in solving polynomial equations.

In summary, the Remainder Theorem and the Factor Theorem are important tools in understanding polynomial functions and solving polynomial equations. By understanding these concepts, you will be able to apply them in solving various problems in algebra. I suggest practicing these concepts with different examples to gain a better understanding of them.