# 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; 