# 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;

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