# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (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 4, and that when is divided by the remainder is 12.

i.
Find the values of and .

ii.       When a and b have these values, find the quotient and remainder when is divided by (x2 2).

Solution

i.

We are given that; ,

We are also given that when is divided by the remainder is 4.

When a polynomial, , is divided by , the remainder is the constant  We can write the divisor in standard form as; Therefore;     We are also given that when is divided by the remainder is 12.

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 the quotient and remainder when is divided by (x2 2).

We are given that; We have found in (i) that and therefore; We are required to find the quotient and remainder when is divided by (x2 2).

We divide by by (x2 2). Therefore, quotient is 2x-4 and remainder is -2.