## And you thought Quadratics were bad…

So, we’ve gone through two and a bit chapters of Mathematical methods so far, and you are beginning to think that you’ve done the worst of it – it can’t get that much harder, can it? Really?

Well, welcome to cubics, quartics and higher order polynomials – they are the big brothers and sisters to the humble quadratic, and they’d like to have a *word* with you about beating up on their sibling, the poor little quadratic – something about “completing **your** square”

On a more serious note, if you are feeling alright about algebra so far, the next section isn’t any harder – just a few more ideas – nothing new or harder than we’ve done so far, just more. Just like when we started quadratics, the first thing we do with cubics and quartics is to identify the common pattern, and be able to factorise them directly from recognising that pattern.

Once we can do that, we will look at how you can start to graph these functions from their intercepts – but that means we need a way of finding all the intercepts. We already knew from our studies of the Null Factor Law (NFL) that the factors of a quadratic equation can give the intercept. The Remainder theorem (right now, that irritating chorus should be going through your head) allows us to find the factors by finding values for **a** where f(a) = 0 for a given f(x). Combining all this knowledge gives us the ability to sketch almost any graph!

But back to basics – just like there are factoring rules for the difference of perfect squares, there are rules for factoring the sum and difference of cubes:

*a*^{3} + *b*^{3} = (*a* + *b*)(*a*^{2} – *ab* + *b*^{2})

*a*^{3} – *b*^{3} = (*a* – *b*)(*a*^{2} + *ab* + *b*^{2})

You need to memorise these and make sure that you can use them – knowing them and using them will make it easier for you! You also need to be skilled with the polynomial long division process – there is a prior post on this blog (link) that will help you with this (there may be other posts that help for those who look for them…).

The new skill you must master is using your ability to find factors to plot the graphs of higher order polynomials. The interesting thing is what happens when you have multiple identical factors – i.e when you factorise a polynomial and you are left with something like this:

f(x) = (x-a) (x-a) (x-b)

which can be rewritten as:

f(x) = (x-a)²(x-b)^{
}

If you have an x-intercept when you have one factor – what happens when you have a double factor? What about if you have a triple factor? Well, it is actually quite simple, when you think about it – remember that a single factor is an x-intercept – a point where the function passes through the x-axis, so it should be obvious that at a double factor, the function passes through the x-axis twice at that point.

Right now, you’re probably a bit confused – how can you have two x-intercepts in the same place – how can the line of a function pass through the x-axis twice at the same point? Well, as I said – it is simple – it simply “bounces” off the x-axis (goes through downwards, and back upwards at the same point), sort of like the vertex of a parabola. When you have three in one place, the function line would pass through three times – imagine it like an unfolded parabola vertex. Enough talking – watch a video:

Finally, here is the checklist: Book Check 3 – Cubic & Quartic Functions

See you in class!

**Explore posts in the same categories:**Algebra, Mathematics, Year 11

May 6, 2013 at 7:30 PM

Hi Sir, can you please send me a copy of the turtle race sheet (the one that you handed out today)! Thanks !!

May 7, 2013 at 4:27 PM

Hi Arina,

It’s on the weebly (11methods2013.weebly.com). If you’ve forgotten the password, ask me in class.