WEBVTT

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Hi I'm Fiona, this is TMUA 2018,
paper one, and question 5.

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We're told that the function F is defined
by F (X) = X cubed plus AX squared plus

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BX plus C, where A,
B and C take the values 1, 2 and 3,

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with no two of them being equal and not
necessarily in that order.

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We're also told the remainder when F(X)
is divided by X + 2 is R and the

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remainder when F(X) is divided by X + 3
is S.

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And we're asked 'what's the largest
possible value of r - s' and here are the

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options for which from which we can
choose once we've gotten through the

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question.

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OK,
so one of the results that comes up quite

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often in similar questions is the factor
theorem,

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which tells us that if I have a
polynomial function F(X) and I know there

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is a factor X minus alpha,
let's say then F(alpha) equals 0.

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Well,
there is a related result called the

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remainder theorem,
where let's say I had X minus beta and I

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knew that the remainder,
when dividing F(X) by X minus beta is

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going to be R,
So non zero then F(beta) equals R.

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So I'm going to use this result here to
answer this question.

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And you might want to take a moment to
pause the video to use what I've just

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said to go through the question yourself,

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or you can carry on following along and
we'll answer the question together.

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So we want to find an expression for r -
s and then make it as large as possible

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using the values 1, 2 and 3 for A,
B and C.

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And we're given two statements here.

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I'm going to label these S1 and S2 so I
can refer back to them using the

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remainder theorem.

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S1 tells me that F(-2) = r,
so that will give me an expression for R,

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so let's calculate that

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then -2 substituted in here for X,
I get R equals -8 plus 4A -2 B plus C.

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So there's an expression for R.
I'm going to use S2 now to find an

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expression for S.
Now I'm going to bring these two

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expressions together to find an
expression for r - s.

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So r - s = - 8 minus - 27,

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that's going to be a +19. 4a - 9a,
which is going to be a - 5a minus 2b

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minus - 3b,
which is going to be a positive 3B.

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So that gives me positive B and then C -
C, which just gives me 0.

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So here's my expression for R - S.
I'm going to use the values 1,

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2 and 3 and allocate them each to A,
B and C in order to make this as large as

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possible.

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So I can see that this middle term here
is going to be negative because A,B...

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the only values I have to choose to
allocate to A, B and C are positive,

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and so this term is going to be negative.

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I want to make that as small as possible,
so I'm going to choose a = 1.

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This term here is positive and I'm going
to make that as large as possible.

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So I'm going to choose b = 3.

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Now you'll see that my expression for r -
s doesn't even include C,

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so it doesn't matter what C is.

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But here C would be allocated to be a
value of 2.

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But now I can just calculate my value,
my largest value for r -s,

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it's going to be 19 - 5 * 1, which is 5,
+ 3

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and so that's going to give me a value of
17.

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And therefore my answer is D17.

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Let's take a brief moment to reflect back
on this question.

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If you weren't confident with application
of the factor theorem and the remainder

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theorem,
you might spend a whole load of time

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dividing this F(X) by X + 2 and by X + 3
to get expressions for R&S.

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You would still arrive at the same answer.

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But it's so important for you to be very
fluent in your application of the factor

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theorem and the remainder theorem,
and this question is a good example of

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why.

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That saves you a lot of time and energy
and can be quite an efficient question to

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answer in the in the exam.