# Polynomials

**Single Variable Polynomials**

"I hate variables. Why are you making this so
complicated?" is a question

I hear while listening in on classes teaching polynomials. The answer is

not very difficult: polynomials are complicated. One of the annoying things

about polynomials is that there are a lot of terms that need to be understood

in order to be able to follow the lecture.

Before actually defining a polynomial, lets take a step
back to simple

expressions. If I ask: "What is the coefficient, and what is the variable

in the expression 5x," the perfect class would unanimously respond "the

coefficient is 5 and the variable is x." These are fundamental concepts in

algebra, and the difference between a variable and a coefficient needs to be

fully understood before trying to understand polynomials.

Next, if I write "9, 11, 5, 2, 15" on the board and ask
the class to tell me

what they are in descending order, they would say: "15, 11, 9, 5, 2." This is

because descending means biggest to smallest.

If I were then to ask how many "terms" were in the
expression 5x + 3, I

would hope to get a response of 2. This is the same "terms" as in \combine

like terms." Now, if I write 5x^{2} + 2x - 3 on the board and ask how many

terms are in this expression, they would hopefully say 3.

Finally, it is time for a definition of a polynomial. A
polynomial in one

variable is an expression of the form:

where are real numbers, and n is a non-negative integer.

It is time to use some specific terms with respect to this
polynomial.

First, the polynomial above is said to be a polynomial of the variable x (or

a polynomial in x) because x is the variable in the expression. The terms

are called the coefficients, with
having the special name of

leading coefficient because it is in front, and
is the constant term

because it is just some real number and does not change. We have written

the expression in descending order with respect to the exponent of x.

Finally, the degree of the polynomial is just the highest power of x. In the

expression above, the degree is n.

**Polynomials of Several Variables**

Polynomials in several variables are what they sound like
{ polynomials with

more than one variable. What does this look like? This is best explained

with an example:

This is a polynomial in two variables, a and b. Like
single-variable polynomials,

these terms have a degree also. However, the degree of a term in a

polynomial in several variables is the sum of the exponents of the variables

in a term. Finally, the degree of the polynomial is the degree of the term

with the highest degree, just like in single-variable polynomials.

Right now, it is enough to know the terminology that goes
with these

polynomials. The leading coefficient is defined the same as it is for polynomials

of one variable. We will leave these now and move on to manipulating

polynomials.

**Adding and Subtracting Polynomials**

Adding and subtracting polynomials is not very different
than manipulating

any other algebraic expression. We add and subtract plynomials by combining

like terms. Combining like terms involves collecting terms with variables

of the same degree, and then combining them by adding or subtracting them

as indicated. For example, 5x^{3} and 2x^{3} are like terms because they have the

same variable and the same exponent. Suppose we have 5x^{3} - 2x^{3}. We can

rewrite this as 5x^{3} +(-2)x^{3}. Now, we can use the distributive law in reverse

to write this as (5+(-2))x^{3}. Then, we can perform the operation inside the

parenthases in accordance with the order of operations to get 3x^{3}.

To make this more concrete, a complete example is needed.
Suppose we

want to add two polynomials: x^{3} +4x^{2} -1 and 2x^{4} +3x^{3} -2x^{2} +x-3. This

addition would be written as:

(x^{3} + 4x^{2} - 1) + (2x^{4} + 3x^{3} - 2x^{2} + x - 3)

Since there is nothing that can be done to simplify the
expressions inside the

parenthases, we can get rid of them. We now have

x^{3} + 4x^{2} -1 + 2x^{4} + 3x^{3} - 2x^{2} + x - 3

Now, we can rewrite this long expression (which is a new
polynomial) in

descending order:

2x^{4} + x^{3} + 3x^{3} + 4x^{2} - 2x^{2} + x - 1 + (-3)

Now, we use the distributive law in reverse to get

2x^{4} + (1 + 3)x^{3} + (4 - 2)x^{2} + x + ((-1) + (-3))

Evaluating these coe cients yields:

2x^{4} + 4x^{3} + 2x^{2} + x - 4

At this point, we have no more like-terms to combine, so
we are done. So,

this is the sum of the above two polynomials.

**Multiplying Polynomials**

Multiplication of polynomials can get complicated.
However, it is nothing

more than the distributive property applied many times. The net result is

that every term in the first polynomial is multiplied by every term in the

second polynomial. To see this, we will use an example. Suppose we want to

multiply x^{3} + 4 and 2x^{2} + x. We can write this as

(x^{3} + 4)(2x^{2} + x)

Now, we can use the distributive law to get:

2x^{2}(x^{3} + 4) + x(x^{3} + 4)

Then, we must distribute two more times:

2x^{2} ∙ x^{3} + 2x^{2} ∙ 4 + x ∙ x^{3} + x ∙ 4

Doing the multiplication gives us:

2x^{5} + 8x^{2} + x4 + 4x

Since there are no like terms to combine, we fininsh by
writing this in descending

order:

2x^{5} + x^{4} + 8x^{2} + 4x

Now, a little more complicated example is:

(x^{3} + 2x^{2} - x - 1)(x - 1)

We have to distribute.

x(x^{3} + 2x^{2} - x - 1) + (-1)(x^{3} + 2x^{2} - x - 1)

Doing the two distributions, we get:

(x^{4} + 2x^{3} - x^{2} - x) + ((-1)x^{3}
+ (-2)x^{2} + x + 1)

Next, we write this in descending order:

x^{4} + 2x^{3} - x^{3} + (-1)x^{2} - 2x^{2} - x + x + 1

Our next task is to undo the distribution of the coefficients for like terms:

x^{4} + (2 - 1)x^{3} + ((-1) + (-2))x^{2} + ((-1) + 1)x + 1

Doing the additions inside of the parenthases yields

x^{4} + 2x^{3} + (-3)x^{2} + 0x + 1

Which is equivalent to

x^{4} + 2x^{3} - 3x^{2} + 1

And we are done!