Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2015  OctNov  (P29709/22)  Q#4
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Question
The polynomial is defined by
where is a constant. It is given that is a factor of
i. Use the factor theorem to show that .
ii. When ;
a. Factorise p(x) completely,
b. Solve the equation for .
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 the factor in standard form as;
Therefore;
ii.
a.
We are given that;
We have found in (i) that therefore;
We are required to factorise completely.
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;
b.
We are required to solve the equation;
Let , then;
We have found in (i) that therefore;
We have also found above that;
Therefore;
Now we have three options.









Since ;



provided that






We know that;
Therefore;


NOT 
Therefore, only possible solution is;
Using calculator;
To find the other solution of we utilize the odd/even property of .
Properties of 

Domain 

Range 

Periodicity 



Odd/Even 

Translation/ Symmetry 






We use odd/even property;
Therefore, we have two solutions (roots) of the equation;


To find all the solutions (roots) over the interval , we utilize the periodic property of for both these values of .
Therefore;
It is evident that all solutions obtained through use of periodic property will be beyond desired interval, therefore, only possible solution within the desired interval is.
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