f(a) (aa)h(a) 0? f(a) 0.
Hence, any polynomial f(x) will be divisible by (xa), if and only if f(a) 0.
This formula is known as factorisation theorem or factor theorem.
Corollary : If az 0 , the polynomial axb will be a factor of any polynomial
f(x), if and only if ̧ 0
¹·
̈
©§
ab
f.Proof :az 0 ,axb= ̧
¹
·
̈
©§
ab
ax will be a factor of f(x), if and only if ̧
¹·
̈
©§
ab
x =̧
¹·
̈
©§
ab
x is a factor of f(x), i.e. if and only if ̧ 0
¹·
̈
©§
ab
f. This method ofdetermining the factors of polynomial with the help of the remainder theorem is also
called the Vanishing method.
Example 1. Resolve into factors : x^3 x 6.
Solution : Here, f(x) x^3 x 6 is a polynomial. The factors of the constant 6
arer 1, r 2,r 3 and r 6.
Putting, x 1 , 1 , we see that the value of f(x) is not zero.
But putting x 2 , we see that the value of f(x) is zero.
i.e., f( 2 ) 23 2 6 8 2 6 0
Hence,x 2 is a factor of f(x)
? f(x) = x^3 x 6
= x^3 2 x^2 2 x^2 4 x 3 x 6
= x^2 (x 2 ) 2 x(x 2 ) 3 (x 2 )
= (x 2 )(x^2 2 x 3 )
Example 2. Resolve into factors : x^3 3 xy^2 2 y^3.
Solution: Here, consider x a variable and y a constant.
We consider the given expression a polynomial of x.
Let, f(x) x^3 3 xy^2 2 y^3
Then, f(y) y^3 3 yy^2 2 y^3 3 y^3 3 y^3 0
?(xy) is a factor of f(x).
Now,x^3 3 xy^2 2 y^3
= x^3 x^2 yx^2 yxy^2 2 xy^2 2 y^3Again let, g(x) x^2 xy 2 y^2
?g(y) y^2 y^2 2 y^2 0