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