Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | May-Jun | (P2-9709/21) | Q#6

Hits: 18

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.

Please follow and like us:

Comments