# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | Oct-Nov | (P2-9709/02) | Q#3

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**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.

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