Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | Oct-Nov | (P2-9709/02) | Q#3
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Question
The polynomial is denoted by
.
i. It is given that is a factor of
. Find the value of
.
ii. When has this value, verify that
is also a factor of
and hence factorise
completely.
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 factor in standard form as;
Therefore;
ii.
We are required to verify that is also a factor of
.
We are given that;
We have found in (i) that , therefore;
When a polynomial, , is divided by
, and the remainder is ZERO i.e.
, then
is factor of
.
Therefore;
Hence is a factor of
.
Therefore, now we have two factors of which are
and
.
A (non-zero) polynomial, , is an expression in
of the form
Where are real numbers,
and
is a non-negative integer.
Number is called degree of the polynomial.
It is evident that is a degree 4 polynomial and hence will have 04 linear factors or 02 quadratic
factors.
We can obtain one of the quadratic factors by multiplication of already obtained 02 linear factors.
Hence, we have quadratic factor of as
.
When a polynomial, , is divided by
, and
is factor of
, then the remainder is ZERO i.e.
.
Therefore, if we divide of by
we will get another quadratic factor of
with ZERO remainder.
Therefore;
The second quadratic factor cannot be further factorized.
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