Polynomials are functions of the form
Some quick terminology; the leading term is the term with the highest power. The leading coefficient is the coefficient of this term. Degree of a polynomial is it's highest power. Remember, all powers in polynomials must be integers greater than or equal to zero.
The theorem to remember here is the remainder theorem (which is the general case of the factor theorem).
If p(x) is a polynomial, the remainder when the polynomial is divided by (x-a) is equal to p(a). Or, in other words:
where Q(x) is a polynomial of lower degree than the original.
This often links to advanced factorisation questions, or locating missing values:
Example (HSC 2014): The polynomial below has a factor x – 2. What is the value of k? We apply the factor theorem to deduce that P(2), so:
You also have to know the more generalised cases of the root equations above.
The above is for a cubic equation. It is rare to get a quartic, but nothing prohibits it, so it might be worth some practice!
Next, estimation of roots. In the interest of brevity, I won't go through an example, but there are two methods you should know:
The Bisection Method involves taking two points either side of a root. One will have a positive sign, the other a negative. We can repeatedly bisect the interval between the two points, checking the sign as we do so, to reach more accurate estimations!
Newton's Method involves the derivative, and is to with the tangent of the curve at a chosen point. The formula for the more accurate root, given a point, is:
Such questions are normally simple substitution style questions for easy marks. However, they will occasionally test you by asking which is a better method. The answer is, almost always, Newton's method. HOWEVER, if a turning point exists between the root and the test point, Newton's method fails.
That is asked repeatedly, so don't forget that! The final area of interest for extension students is inverse functions. An inverse function can be considered as a sort of reverse function, which takes the outputs of the original and gives back the original inputs. Now, questions on inverse functions specifically (not concerning the inverse trig functions) are rare, so again, in the interest of brevity, I will skip an example (I don't want you having to read through massive documents instead of studying, these are meant to be quick reads ;)). However, if you have a specific question, post it below, and I will happily walk everyone through it! Here are the main things you should know about inverse functions:
- Inverse functions exist only if each output can only be achieved through a single input. You likely know this as the horizontal line test, but technically, it means the function is what is called one-to-one.
- The inverse of a function can be found be reversing the positions of the y and the x, and rearranging.
- The inverse of a function reverses the domain and range. That is, the range becomes the domain and the domain becomes the range
- A function is symmetrical with it's inverse about the line y=x.
Again, detailed questions on inverse functions are rare. It usually sticks to finding an inverse and/or graphing it. But be sure to practice them, because lots of weird and wonderful things can be asked if BOSTES feels particularly nasty. Be sure to post any questions below and I will run through them for everyone to benefit from.