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

Hits: 20

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 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;    