# 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;