Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2018  MayJun  (P29709/21)  Q#6
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Question
The cubic polynomial is defined by
where is a constant. It is given that is a factor of
i. Use the factor theorem to find the value of and hence factorise f(x) completely.
ii. Hence, without using a calculator, solve the equation f(2x) = 3f(x).
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;
We are given that;
We are also given that is a factor of .
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;
ii.
We are required to solve the equation f(2x) = 3f(x) without using a calculator.
We are given that;
We have found in (i) that therefore;
We need to factories this polynomial.
First we find a linear factor of the polynomial.
If is a factor of then according to factor theorem, however, here will be a factor of and will be a factor of .
We can substitute all these possible values for , one by one, in the given and find the linear factor. Although a careful observation of known and potential factors may point to some values more probable than others, however, the method remains strictly “trial and error”.
We can start from ;
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;
Hence is not a factor of .
Next, we can check ;
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;
Hence is a 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;
Now we have three options.






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