# 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 (x^{2 }− 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 (x^{2} − 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 (x^{2} − 2).

We divide by by (x^{2 }− 2).

Therefore, quotient is 2x-4 and remainder is -2.

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