Question

The polynomial  is denoted by .

    i.       It is given that  is a factor of . Find the value of .

 ii.   When  has this value, verify that  is also a factor of  and hence factorise  completely.

Solution

i. 

We are given that;

We are also given that  is a factor of .

When a polynomial, , is divided by , and  is factor of , then the remainder is ZERO i.e. .

We can write factor in standard form as;

Therefore;

 ii.
 

We are required to verify that  is also a factor of .

We are given that;

We have found in (i) that , therefore;

When a polynomial, , is divided by , and the remainder is ZERO i.e. , then  is factor of .

Therefore;

Hence  is a factor of .

Therefore, now we have two factors of  which are  and .

A (non-zero) polynomial, , is an expression in  of the form

Where are real numbers,  and  is a non-negative integer.

Number  is called degree of the polynomial.

It is evident that  is a degree 4 polynomial and hence will have 04 linear factors or 02 quadratic
factors.

We can obtain one of the quadratic factors by multiplication of already obtained 02 linear factors. 

Hence, we have quadratic factor of  as .

When a polynomial, , is divided by , and  is factor of , then the remainder is ZERO i.e. .

Therefore, if we divide of  by  we will get another quadratic factor of  with ZERO  remainder.

Therefore;

The second quadratic factor  cannot be further factorized.