Hi, recently I was given a question by a teacher:
Simplify:
After the whole class struggled over the week, our teacher revealed the answer:
How do I get to this answer??
Please help, thanks BG
Hello BGYour only hope would be to find the factors of the numerator and denominator - and then 'cancel' the common factor, which turns out to be ; i.e.
The only way I can suggest you might have done this is to use a 'trial and error' process on each of the quartic expressions. Given that, for example, cannot be broken down in many ways -
- ; and
- this shouldn't have taken too long.
But I must admit: I cheated and worked back from the answer you gave me using algebraic long division.
Does that help to shed some light?
Grandad
Hello reiwardYou haven't quoted artvandalay11's posting quite accurately. The expressions factorise as follows:
and then the common factor is 'cancelled'.
So how do we find these factors? If you look back to my first posting, you'll see that I've given some clues, by looking at the possible factors of . The process is not dissimilar to factorising quadratics - just a bit more complicated!
Any factors (if they exist at all) of a quartic expression in will be either in the form:
- A linear factor and a cubic factor, like , where the cubic may or may not factorise further; or
- Two quadratic factors, like , where neither quadratic factorises further.
Looking back to my posting, then, you have three possibilities for :
- Linear and cubic:
- Linear and cubic:
- Two quadratics:
Then you'd look at the factors of the constant term, , which of course are simply , and, by trial and error eliminate the first two of these three possibilities, by considering the coefficients of and .
This would leave you with either or , where and are constants still to be found. Again, by looking at the coefficients of and you should fairly quickly see that the factors are .
The denominator can be factorised in a similar way, except that by now you will have found the factors of the numerator, and you'll be hoping that one of these will also be a factor of the denominator in order to allow some simplification to take place. So this gives quite a helpful clue as to what the factors are likely to be!
Not easy, but definitely achievable!
Grandad