Polynomials in Mathematics

In this post, I would like to briefly refresh the concept of Polynomials in Mathematics, as at the start of the school year there are always students looking for extra polynomials help.

I would like to explain some basic terms and definitions that will hopefully make working with polynomials less confusing.

The first two terms which are fairly obvious are "variable" and "constant".  You probably don't need me to tell you that constants are terms that do not change (they are fixed... they don't always have to be known, but they do not change), whereas you can change the value of the variable to arrive at a different solution to your problem.  As in, if you vary the length of the square, you calculate a different area of the square.  The general practice is to assign letters near the start of the alphabet as constants (a, b, c), and at the end for variables (x, y, z).  These terms are self-explanatory, so I will leave them at that.

Substituting values into your variables only makes sense in some cases.  E.g.  sub in -4 for x, when the expression is square root of x.  It doesn't make sense.  Therefore, the set of numbers that DO make the expression make sense is called the "domain of the variable".

An example of this concept can be seen here.  Find the domain of the expression 1 / (x-5).

You can substitute any number in for x and calculate a solution, except for one.  You cannot have a zero as the denominator, as that makes the expression undefined.  So, in this case, you have to ask yourself "what makes the denominator equal zero?"  You can calculate in this case that x - 5 = 0, and so x=5.  Therefore, the domain of the original expression is all real numbers, except x cannot equal 5.

The term "polynomial" is given when you have a series of "terms" in an expression with different powers of x as long as the powers are not negative.  In general, a polynomial looks like this:

a_nxⁿ + a_n-1x^n-1 + ... a₁x + a₀

Each x value is contained in a "term".  The a values are "coefficients", which are constants.  They do not vary.  Overall, the "degree" of the polynomial is the highest power of x.  General convention is to right the terms from the highest degree of x to the lowest.

Since a polynomial is a generic collection of many (poly = many) terms, specific forms of polynomials have special names.  A "monomial" is a polynomial that only has one term, such as 2x.  A "binomial" has two terms, such as 4x² - 1x.  And a "trinomial" has three terms (guess what it looks like!).

Polynomial terms of the same degree can be combined.

In the expression 3x + 2x + 1, you can combine the terms so that you simplify to 5x + 1.  You can do this with x² terms and higher as well.

There are a couple of special polynomial products that will appear very frequently in your studies.  I will also go over these in my next post so that you can memorize them to apply them easily.  Hopefully, when you see these, you will understand how they work and will not need to look for polynomials help anymore!

Radicals and Rational Exponents

Having discussed the topic of radicals and nth roots in my previous set of posts, we can apply that information to the concept of fractional exponents. These rational exponents are no more difficult to work with, and once you identify just what they are telling you, you can easily apply your rules for exponents to simplify them and solve your equations. Factoring fractional exponents may look tricky, but you will see that they follow the same rules that you already know.

Consider the expression 2^1/3. Knowing your rules for exponents, you can see that if we cube this so that we get (2^1/3)³, this is equal to 2¹, which is just 2. However, what you will realize is that what we have just done is apply property #1 of the properties of nth roots. That is, (ⁿ√x)ⁿ = x. So, what we have just shown is that 2^1/3 is the same as ³√2.

So then, with that example we can now extend what we've done to come to a definition of rational exponents and roots.

Part 1:  If we let b be any real number, and n is a natural number, b^1/n can be defined by ⁿ√b. (if n is even, b must be greater than or equal to 0).

Furthermore, we can extend our definition of exponents fractions to include any fraction, not just 1/x type.

Part 2:  If we let m/n be a rational number, n is positive, and ⁿ√b exists, we can say that b^m/n = ⁿ√(b)^m or (ⁿ√b)^m. (Both forms say the same thing and work the same way.)

It is important to realize and understand that these rational exponents follow the same properties as integer exponents, which you are likely already familiar with (I'll do a post soon to refresh you, or for those who have not learned that topic yet).

So now that you know what rational exponents are, let's take a look at a few examples of how you work with radical and rational exponents.  Simplifying rational exponents takes some practice, but it is all the same math that you have already learned.

1.  Simplify the following expressions:
a) 64^1/2
b) -64^1/2
c) (-64)^1/2
d) 64^-1/2

a) This is simply the square root of 64, √64, which is 8.
b) Recall the notation with negative signs and nth roots. This means -(64^1/2) = -√64 = -8.
c) As there is no such real number that gives -64 when squared, this expression cannot be simplified to a real number.
d) 64^-1/2 indicates √64^-1, and a negative exponent means that you take the inverse of it. So,64^-1 means 1/64, and then the square root of this is 1/8.

2.  Simplify (4x^2/3)(3x^1/4)
(4x^2/3)(3x^1/4) = 12x^{(2/3) + (1/4)} (by the exponent rules)
= 12x^11/12

3.  Simplify (√x)(³√y⁴)
= (x^1/2)(y^3/4)
Alternatively, if we give the fractions a common denominator, we can rewrite as follows:
(x^1/2)(y^3/4) = (x^2/4)(y^3/4) = (⁴√x²)(⁴√y³) = ⁴√(x²y³)

So, there you have it.  Working with radicals and rational exponents is not that different from working with any other exponents, once you know what you're doing.  Go over my examples again to be sure you understand the concepts, and practice some more on your own, and you will be an expert in no time. Drop me a line in the comments if you have any problems.

One final note: irrational or radical exponents follow the same rules as well, and I will cover some of those examples in a future post.