Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2010  OctNov  (P29709/23)  Q#3
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Question
The polynomial x^{3} + 4x^{2} + ax + 2, where a is a constant, is denoted by p(x). It is given that the remainder when p(x) is divided by (x + 1) is equal to the remainder when p(x) is divided by (x − 2).
i. Find the value of a.
ii. When a has this value, show that (x − 1) is a factor of p(x) and find the quotient when p(x) is divided by (x − 1).
Solution
i.
We are given that;
We are also given that the remainder when is divided by the remainder is equal to the remainder when is divided by .
When a polynomial, , is divided by , the remainder is the constant
We can write both divisors in standard form as;


According to the given condition;
ii.
We are required to show that when , (x1) is the factor of the polynomial p(x).
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;
Hence (x – 1) is a factor of the polynomial p(x).
Next we are required to find the quotient when p(x) is divided by (x – 1). We perform long division.
Hence the quotient is .
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