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Factorization of Polynomials

If polynomial is of degree

n³ 1 and a is any real number, then

(i) x- a is a factor of , if p(a)=0 and

(ii) p(a)= 0, if x- a is a factor of .


Proof : By the Remainder Theorem,

(i) If p(a)=0, then , which shows is a factor of

(ii) Since is a factor of , for some polynomial .

= 0

 

Example :

Check whether is a factor of

The zero of is

Substitute in

                                =

                                = 8 - 12 + 8 - 4

                                = 16 - 16

                                = 0

Since =0, is a factor of

 

Example : Find the value of if is a factor of =

Given is a factor of =

                                   

                          

                              

                                

                                     

                                      

Factorization of the polynomial by splitting the middle term:

Factorize in to the factors of the form and

Let =

                  =

Comparing on both sides we have, and

If we find and satisfying the above two conditions, we can write,

 

=

            

           

           

           

 

Example : Factorize splitting the middle term

Here

and

Here 8 = 8 ´ 1 but 8 + 1 ≠ 6

 

       8 = 4 ´ 2 and 4 + 2 = 6

Therefore split the middle term as and

=

             =

             =

The factors of are





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