Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/23) | Q#3

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Question

The polynomial x3 + 4x2 + 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 , (x-1) 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 .