# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/22) | Q#7

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Question

The polynomial , where and are constants, is denoted by . It is given that is a factor of , and that when is divided by the remainder is 10.

i.
Find the values of a and b.

ii.       When a and b have these values, solve the equation p(x)=0.

Solution

i.

We are given that; We are also given that is a factor of .

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

We can write factor in standard form as; Therefore;    We are also given that when is divided by the remainder is 10.

When a polynomial, , is divided by , the remainder is the constant  We can write 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 solve .

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

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

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

We divide by . Therefore;      Since we are given that;            