# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2014 | June | Q#5

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**Question**

The polynomial is given by

.

where c and d are integers.

**a. ** Given that is a factor of show that

**b. **The remainder when is divided by is 65. Obtain a function in c and d.

**c. **Use equations from parts (a) and (b) to find the value of c and the value of d.

**Solution**

**a.
**

Factor theorem states that if is a factor of then;

For the given case is factor of .

We can write the divisor in standard form as;

Here and . Hence;

**b.
**

Remainder theorem states that if is divided by then;

For the given case is divided by and remainder is 65.

Here and . Hence;

**c. **

We are required to find values of c and d from equations obtained in parts (a) and (b).

These equations are;

From both equations we can find expressions for ‘d’.

Equating both expressions of ‘d’ we can find the value of ‘c’.

Substituting in any of the equations obtained in parts (a) and (b).

We choose;

Hence;

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