Hey it’s Professor Dave, let’s learn about polynomials.
Now that we know the very basics of algebra and algebraic equations, it’s time to tackle
a particular type of equation that you will see countless times.
These are called polynomials, and they are expressions that contain an unknown raised
to various positive whole number exponents.
This could be something like two X squared plus four X plus ten equals zero.
We can see that there is an X squared term, an X term, and a number or constant, and that’s
what makes this a polynomial, because it follows the general form of (A sub N) (X to the N),
plus (A sub N minus one) (X to the N minus one), plus (A sub N minus two) (X to the N
minus two), and so on until we get to (A one) X, plus A zero, a constant.
If there is just one term, like two X, it’s a monomial.
If there are two terms, like two X minus one, it’s a binomial.
Three terms, like X squared plus two X minus one, would be a trinomial.
Beyond this there are no special names, so we can generally just call all of these polynomials,
whether there is one term or a hundred.
We definitely could have polynomials with many terms, going up to any integer exponent
there could be, though we typically only interact with expressions that go up to a power of
three or four, maybe five or six once in a while.
As we referenced, it is convention to list the terms in the polynomial by decreasing
exponents, so we would say three X cubed plus five X squared minus two X plus seven equals
zero, in that specific order, and typically not in some other order.
Again, there is no firm reason for this, it’s just convention.
We should also note that the highest power present in the polynomial is called the degree
of the polynomial.
If the degree is two, it is a second degree polynomial, also known as a quadratic polynomial.
Third degree is known as cubic.
Fourth degree is called quartic.
Fifth degree is called quintic, and beyond that we don’t really have names, we would
just say sixth degree, seventh degree, and so on.
Also, polynomials don’t have to have all the possible terms, four X squared plus seven
is a polynomial even without a term for X raised to the first power, we can just think
of it as zero X, which we simply omit.
If we want to evaluate a polynomial for a specific value of X, it’s pretty easy, we
just have to plug in that value everywhere we find X.
Using the previous expression, two X squared plus four X plus ten, if we want to evaluate
this when X equals negative one, we just plug it in.
X squared will be one so we get two for the first term, four X will end up as negative
four, and then the plus ten, so two minus four plus ten, which is eight.
Now solving polynomials, on the other hand, can be extremely complicated.
Let’s take the exact same expression we just looked at and set it equal to zero, as
we typically do with polynomials.
Now that we have an equation, we might want to solve for X, meaning we may want to find
the values of X for which this expression would indeed equal zero.
But we will quickly notice that all the strategies we have learned thus far will not work, because
we have more than one X term, and they are not like terms that can be combined.
We could subtract ten from both sides, but then we are pretty much stuck.
If we take a square root to try to lose the exponent, we get roots involved, so it would
seem that we have no course available to solve for X.
But don’t worry, there are indeed a variety of ways to solve these kinds of equations,
and we will learn them all one by one, but it is worthwhile to note that when polynomials
get much more complex than this, their solutions become so intricate that even brilliant mathematicians
spend a lifetime trying to figure out ways to solve them.
We won’t be doing any of those, of course, we will just focus on the basics, but it is
a good idea to show some reverence for the deep complexity that algebra is capable of.
For now just put that reverence in your pocket, as we’ve got a lot more work to do with
polynomials, so let’s move forward now.