# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/23) | Q#7

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Question

The polynomial , where a and b are constants, is denoted by . It is given that when is divided by the remainder is 30, and that when is divided by the remainder is 18.

i.
Find the values of and .

ii.       When a and b have these values, verify that (x 2) is a factor of p(x) and hence factorise completely.

Solution

i.

We are given that; We are also given that when is divided by the remainder is 30.

When a polynomial, , is divided by , the remainder is the constant  Therefore;       We are also given that when is divided by the remainder is 18.

When a polynomial, , is divided by , the remainder is the constant  We can write the divisor in standard form as; Therefore;     From we can substitute in above equation ;    Substitution of in any of these two equations yields value of . We choose;    ii.

We are required to find other linear factors of .

We are given that; We have found in (i) that and therefore; We are required to show that is factor of .

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

Therefore;     Hence, is factor of .

Therefore, division of with factor will yield a quadratic factor with ZERO remainder.

We divide by by . Therefore;      