# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2016 | June | Q#4

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Question

The polynomial  is given by .

a.

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

ii.      Express  as a product of three linear factors.

b.

i.       Use Remainder Theorem  to find the remainder when  is divided by .

ii.       Express  in the form , where b and c are integers.

Solution

a.

i.

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 .

ii.

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 2nd and 3rd 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;

b.

i.

Remainder theorem states that if  is divided by  then;

For the given case   is divided by .

Here  and . Hence;

ii.

From (b:i) we know that when    is divided by  we get 20 as  remainder.

When a polynomial  is divided by divisor  and quotient is  with reminder R, we can express it as;

Therefore, for the given case;

However, to find the quotient , we need to divide   by .

Here, we can see that when   is divided by  we get  as quotient.

Therefore;