# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | May-Jun | (P2-9709/02) | Q#4

Question

The cubic polynomial , where and are constants, is denoted by . It is given that and are factors of .

i.       Find the values of and .

ii.       When and have these values, find the other linear factor of .

Solution

i.

We are given that; We are also given that and are a factors of .

When a polynomial, , is divided by , and is factor of , then the remainder is ZERO i.e. .

We can write the factors in standard form as;  Therefore;            We can substitute from into .    Substitution of in any of these two equations yields value of . We choose;   ii.

We are required to find other linear factor of .

We are given that; We have found in (i) that and therefore; We are also given that and are factors of . Therefore, division of with any  of these factors will yield a quadratic factor with ZERO remainder.

We divide by by . Therefore;     Hence, is another linear factor of .