In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout)[1] is an application of Euclidean division of polynomials. It states that, for every number
r,
f(x)
f(r)
x-r
x
f.
f(r)
f(x)
x-r,
x-r
f(x)
f(r)=0,
Let
f(x)=x3-12x2-42
f(x)
(x-3)
x2-9x-27
-123
f(3)=-123
Proof that the polynomial remainder theorem holds for an arbitrary second degree polynomial
f(x)=ax2+bx+c
So, which is exactly the formula of Euclidean division.
The generalization of this proof to any degree is given below in .
The polynomial remainder theorem follows from the theorem of Euclidean division, which, given two polynomials (the dividend) and (the divisor), asserts the existence (and the uniqueness) of a quotient and a remainder such that
f(x)=Q(x)g(x)+R(x) and R(x)=0 or\deg(R)<\deg(g).
If the divisor is
g(x)=x-r,
f(x)=Q(x)(x-r)+R.
Setting
x=r
f(r)=R.
A constructive proofthat does not involve the existence theorem of Euclidean divisionuses the identity
xk-rk=(x-r)(xk-1+xk-2r+...+xrk-2+rk-1).
Sk
n+a | |
f(x)=a | |
n-1 |
xn-1+ … +a1x+a0,
f(x)-f(r)=(x-r)(anSn+ … +a2S2+a1),
S1=1
Adding
f(r)
The polynomial remainder theorem may be used to evaluate
f(r)
R
The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.[3]