1. Simultaneous equations & equating coefficients:We can equate coefficients here and we get a pair of simultaneous equations:
&
:.
Substituting back in, you get
2. Calculus (or expansion if you like):I assume you meant
, because otherwise this equation doesn't make alot of sense
This is the form of a cubic in point-of-inflection form, so using calculus to find this point:
Let
:
So we have
as a location for our point of inflection:
Substituting x = 3 into original equation:
so we have the pair:
Now remembering our general cubic form:
, where h is the x-coordinate of the point of inflection and k is the y-coordinate. a is the dilation factor (in this case, it is 1, because in the LHS the
term has a coefficient of 1
so
:.
3: Polynomial Long Division: Remainder:
Remebering our form for long division:
Where
,
,
&
:
4. InverseThis function is one-to-one, so it will have an inverse:
To find an inverse, just swap around the x & y variables then rearrange the equation:
Now for this equation to not have a divide-by-zero, you will need the following domain:
Domain:
(which is the range of our original function)