Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2010  OctNov  (P29709/22)  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;










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