# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2010 | January | Q#1

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**Question**

The polynomial is given by .

**a. **Use the Factor Theorem to show that is a factor of .

**b. **Express as the product of three linear factors.

**Solution**

**a.
**

Factor theorem states that if is a factor of then;

For the given case is factor of .

We can write the factor in standard form as;

Here and . Hence;

Hence, is factor of .

**b. **

If is a polynomial of degree then will have exactly factors, some of which may repeat.

We are given a polynomial of degree ,

Therefore, it will have 03 factors some of which may repeat.

From (a:i) we already have 01 factor of .

We may divide the given polynomial with any of the already known linear factor(s) to get a quadratic factor and then factorize the obtained quadratic factor to find the 2^{nd} and 3^{rd} linear factors.

For the given case;

Already known factor from (a:i) is .

We may divide the given polynomial by factor .

Therefore, we get the quadratic factor for given polynomial.

Now we factorize this quadratic factor.

Hence, can be written as product of three linear factors as follows;

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