Past Papers’ Solutions  Assessment & Qualification Alliance (AQA)  AS & A level  Mathematics 6360  Pure Core 1 (6360MPC1)  Year 2010  June  Q#3
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Question
The polynomial is given by
a.
i. Use the Factor Theorem to show that
ii. Express
b. Use the Remainder Theorem to find the remainder when
c.
i. Verify that
ii. ketch the curve with equation
Solution
a.
i.
Factor theorem states that if
For the given case
We can write the factor in standard form as;
Here
Hence,
ii.
If
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
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 2^{nd} and 3^{rd} 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
Now we factorize this quadratic factor.
Hence,
b.
Remainder theorem states that if
For the given case
Here
c.
i.
We are given;
Let’s first find
Now, let’s find
Hence;
ii.
We are required to sketch a cubical polynomial given as;
ü Find the sign of the coefficient of
The coefficient of
ü Find the point where the graph crosses yaxis by finding the value of
Therefore for
Hence, the cubic graph intercepts yaxis at
ü Find the point(s) where the graph crosses the xaxis by finding the value of
From (a:ii) we know that given polynomial can be written as;
It is evident from factors that the function









It is evident cubic graph will intersect xaxis at
ü Calculate the values of
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.
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