Question

Whats The Basic Difference Between Remainder Theorem And Factor Theorem

Answer 1

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REMAINDER THEREOM

In algebra, the polynomial remainder theorem or little Bézout's theorem[1] is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial f ( x ) by a linear x-r is equal to f(x)

FACTOR THEOREOM

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem. The factor theorem states that a polynomial has a factor if and only if (i.e. is a root).

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Answer 2

Basic Proportionality Theorem: If a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides those sides in proportion.

The Remainder Theorem

When we divide f(x) by the simple polynomial x−cwe get:

f(x) = (x−c)·q(x) + r(x)

x−c is degree 1, so r(x) must have degree 0, so it is just some constant r :

f(x) = (x−c)·q(x) + r

Now see what happens when we have x equal to c:

f(c) =(c−c)·q(c) + r

f(c) =(0)·q(c) + r

f(c) =r

So we get this:

The Remainder Theorem:

When we divide a polynomial f(x) by x−c the remainder is f(c)
The Factor Theorem

Now ...

What if we calculate f(c) and it is 0?

... that means the remainder is 0, and ...

... (x−c) must be a factor of the polynomial!

We see this when dividing whole numbers. For example 60 ÷ 20 = 3 with no remainder. So 20 must be a factor of 60.

Example: x2−3x−4

f(4) = (4)2−3(4)−4 = 16−12−4 = 0

so (x−4) must be a factor of x2−3x−4

And so we have:

The Factor Theorem:

When f(c)=0 then x−c is a factor of f(x)

And the other way around, too:

When x−c is a factor of f(x) then f(c)=0


Knowing that x−c is a factor is the same as knowing that c is a root (and vice versa).

The factor "x−c" and the root "c" are the same thing

Know one and we know the other

For one thing, it means that we can quickly check if (x−c) is a factor of the polynomial.