Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2007 | January| Q#1

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Question

The polynomial  is given by

where k is a constant.

a.

i.
Given that  is a factor of , show that .

ii.       Express   as the product of three linear factors.

b.   Use the Remainder Theorem to find the remainder when  is divided by .

c.   Sketch the curve with equation  , indicating the values where the curve  crosses the x-axis and the y-axis. (You are not required to find the coordinates of the stationary points.)

Solution

a.

i.

Factor theorem states that if  is a factor of   then;

We are given a polynomial;

For the given case,  is a factor of .

We can write the factor in standard form as;

Here  and . Hence;

Hence, if  is the factor of  then;

We can write the given polynomial as;

ii.

If  is a polynomial of degree  then  will have exactly  factors, some of which may repeat.

We are given a polynomial of degree ,

Therefore, it will have 03 factors some of which may repeat.

From (a:i) we already have 01 factor  of .

We may divide the given polynomial with any of the already known linear factor(s) to get a quadratic  factor and then factorize the obtained quadratic factor to find the 2nd and 3rd linear  factors.

For the given case;

Already known factor from (a:i) is .

We may divide the given polynomial by factor .

Therefore, we get the quadratic factor  for given polynomial. Now we factorize this quadratic factor.

Hence,  can be written as product of three linear factors as follows;

b.

Remainder theorem states that if  is divided by  then;

For the given case  is divided by .

Here  and . Hence;

When ;

c.

We are required to sketch a cubical polynomial given as;

ü Find the sign of the coefficient of . This gives the shape of the graph at the extremities. If the  coefficient of  is such that  cubic graph is increasing from left to right at the extremities and  when  the cubic graph is inverted.

The coefficient of  in the given polynomial is positive. Therefore it will cause normal cubic graph  increasing from left to right at the extremities.

ü Find the point where the graph crosses y-axis by finding the value of  when .

Therefore for ;

Hence, the cubic graph intercepts y-axis at .

ü Find the point(s) where the graph crosses the x-axis by finding the value of  when . If  there is repeated root the graph will touch the x-axis.

From (a:ii) we know that given polynomial can be written as;

It is evident from factors that the function  will cross/intersect horizontal axis when;

It is evident cubic graph will intersect x-axis at .

ü Calculate the values of  for some value of . The is particularly useful in determining the  quadrant in which the graph might turn close to the y-axis.

ü Complete the sketch of the graph by joining the sections.

Sketch should show the main features of the graph and also, where possible, values where the graph intersects coordinate axes.