.
.
Then multiply out the rest: 
. This can be seen by evaluating f(-1) and using Factor Theorem above. It then remains to see what the quadratic factor will be. Again, by inspection (guessing and multiplying out rather than using polynomial division) we have 
Finally, the quadratic can also be factorised to give 
is divisible by 
.- The easiest way to do this is to use factor theorem. Factor theorem states that f(x) is divisible by x-2 if f(2)=0.
Hence x-2 is a factor of f(x) and f(x) is divisible by x-2. - According to Factor Theorem, if
then
is a factor and we can write
. The quadratic factor is identified by inspection – in order to get
as the first term when expanding, the first term in the quadratic must be
. Since the expansion gives the correct
term we don’t require an x term in the quadratic. It remains to find the constant term which can be seen from the final term in f(x). The quadratic term itself can be factorised as
since once a 2 is factored out, a difference of squares remains. We have
.
Polynomials are linear combinations of powers of x. The highest power is the order of the polynomial. For example, a cubic is a polynomial of order 3.
You may be required to manipulate polynomials including expanding brackets, simplifying expressions and factorising. There may also be questions involving Polynomial Division and/or with the use of Factor Theorem.
Manipulating Polynomials
- Expanding and Simplifying – To expand and simpify
– this works in the same way as expanding double brackets but you should end up with 6 terms before simplification.
- Factorising Cubics – you may be asked to factorise
, for example. The trick is to inspect the cubic and see if you can guess a root. In this case, x=1 is a root since it gives f(1)=0. It follows that x-1 is a factor. It can be shown that
by guessing the quadratic inside the brackets and expanding then improving. See the Cubics page, Example 2.2 for another example. Alternatively, you can use polynomial division as below and seen in Example 3. Fully factorised
.
Factor Theorem & Polynomial Division
- Factor Theorem – Factor theorem states that if a polynomial is divisible by
then
and vice versa. It follows from this that if a polynomial is divisible by
then
and vice versa. For example, if
is a factor of
, then
. Similarly, if
then
is a factor. In simpler cases, if
, for example, then
is a factor of
- Polynomial Division – Polynomial Division combines algebra with the technique of long division. The idea is to identify the factor required for the left most term in each step. This factor then goes on top. For example, in Example 3, a cubic polynomial with a
term is being divided by a linear function with an
term. The missing factor is thus
to see what remains to find.
Examples
Example 3 – Polynomial Division
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