Now, if we indicate the dividend byf(x), the quotient by h(x), the remainder by r
and the divisor by (xa), from the above formula, we get,
f(x) (xa)h(x)r.......... (i) this formula is true to all values of a.Substituting x a in both sides of (i), we get,
f(a) (aa)h(a)r 0 h(a)r r
Hence,r f(a).
Therefore, if f(x) is divided by (xa), the remainder is f(a). This formula is
known as remainder theorem. That is, the remainder theorem gives the remainder
when a polynomial f(x) of positive degree is divided by (xa) without performing
actual division. The degree of the divisor polynomial (xa) is 1. If the divisor is a
factor of the dividend, the remainder will be zero and if it is not zero, the remainder
will be a number other than zero.
Proposition : If the degree of f(x) is positive and az 0 , f(x) is divided by
(axb), remainder is ̧
¹·
̈
©§
ab
f.Proof : Degree of the divisor axb,(az 0 ) is 1.
Hence, we can write,
hx r
ab
fx ax b hx r ax ̧
¹·
̈
©() ( )() § ()? ahx r
a
f x x b
̧
¹·
̈
©() § ()Observe that, if f(x) is divided by ̧
¹
·
̈
©§
a
x b, quotient is ah(x) and remainder is r.Here, divisor = ̧
¹
·
̈
©§
a
x bHence, according to remainder theorem, ̧
¹
̈ ·
©§
ab
r fTherefore, if f(x) is divided by (axb), remainder is ̧
¹
·
̈
©§
ab
f.Corollary :(xa) will be a factor of f(x), if and only if f(a) 0.
Proof : Let, f(a) 0
Therefore, according to remainder theorem, if f(x) is divided by (xa), the
remainder will be zero. That is, (xa) will be a factor of f(x).
Conversely, let, (xa) is a factor of f(x).
Therefore, f(x) (xa)h(x), where h(x) is a polynomial.
Putting x a in both sides, we get,