Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2012 | June | Q#3

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Question

The polynomial is given by .

a.

i.       Use the Factor Theorem to show that is a factor of .

ii.       Express as the product of three linear factors.

b.   Verify that .

c.   Sketch the curve with equation , indicating the values where the curve  crosses the x-axis.

Solution

a.

i.

Factor theorem states that if is a factor of then; For the given case is factor of .

We can write the divisor in standard form as; Here and . Hence;       Hence, is factor of .

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.

We are given that; We are required to verify that;   Hence .

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. 