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Module 1: Polynomial, Rational, and Radical Relationships

A: Introduction to Polynomials

A polynomial is an expression composed of variables, constants, and exponents that are combined using mathematical operations such as addition, subtraction, multiplication, and division (No division operation by a variable).

A polynomial looks like this:

an example of a polynomial with 3 terms

Polynomials with one variable make nice smooth curves:

A polynomial can have:

constants (like 3, −20, or ½)
variables (like x and y)
exponents (like the 2 in y2), but only 0, 1, 2, 3, … etc are allowed
that can be combined using addition, subtraction, multiplication, and division.
A polynomial can have constants, variables, and exponents, but never division by a variable. Also, they can have one or more terms but not an infinite number of terms.

Polynomial or Not?

These are polynomials:
3x
x − 2
−6y2 − (7/9)x
3xyz + 3xy2z − 0.1xz − 200y + 0.5
512v5 + 99w5
5

(Yes, “5” is a polynomial, one term is allowed, and it can be just a constant!)

These are not polynomials
3xy-2 is not, because the exponent is “-2” (exponents can only be 0,1,2,…)
2/(x+2) is not, because dividing by a variable is not allowed
1/x is not either
√x is not, because the exponent is “½” (see fractional exponents)

But these are allowed:
x/2 is allowed, because you can divide by a constant
also 3x/8 for the same reason
√2 is allowed, because it is a constant (= 1.4142…etc)

Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

How do you remember the names? Think cycles!

There is also quadrinomial (4 terms) and quintinomial (5 terms), but those names are not often used.

Variables

Polynomials can have no variable at all.
Example: 21 is a polynomial. It has just one term, which is a constant.
They may have one variable
Example: x4 − 2×2 + x has three terms, but only one variable (x)
Or two or more variables
Example: xy4 − 5x2z has two terms, and three variables (x, y and z)

What is Special About Polynomials?

Because of the strict definition, polynomials are easy to work with.
For example, we know that:
If you add polynomials you get a polynomial
If you multiply polynomials you get a polynomial

So you can do lots of additions and multiplications, and still have a polynomial as the result. Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines.

Example: x4−2×2+x

See how nice and smooth the curve is?

You can also divide polynomials (but the result may not be a polynomial).

Degree

The degree of a polynomial with only one variable is the largest exponent of that variable.

Example:
4×3 − x + 2

The Degree is 3 (the largest exponent of x)

Standard Form

The Standard Form for writing a polynomial is to put the terms with the highest degree first.
Example: Put this in Standard Form: 3×2 − 7 + 4×3 + x6

The highest degree is 6, so that goes first, then 3, 2, and then the constant last:
x6 + 4×3 + 3×2 − 7

Common Symbols Used in Algebra

Topic B: Adding and Subtracting Polynomials

A polynomial looks like this:

example of a polynomial
this one has 3 terms

To add polynomials we simply add any like terms together. Like Terms are terms whose variables (and their exponents, such as the 2 in x2) are the same.

In other words, terms that are “like” each other.

Note: the coefficients (the numbers you multiply by, such as “5” in 5x) can be different.

Example:
7x                   x                   -2x                   πx

are all like terms because the variables are all x

Example:
(1/3)xy2                   -2xy2                   6xy2                   xy2/2

are all like terms because the variables are all xy2

Example: These are NOT like terms because the variables and/or their exponents are different:
2x                   2x2                   2y                   2xy

Adding Polynomials

Step 1: Arrange each polynomial with the term with the highest degree first then in decreasing order of degree.
Step 2: Group the like terms. Like terms are terms whose variables and exponents are the same.
Step 3: Simplify by combining like terms.

Example: Add 2×2 + 6x + 5 and 3×2 – 2x – 1

Start with:
2×2 + 6x + 5 + 3×2 − 2x − 1

Place like terms together:
2×2+3×2 + 6x−2x + 5−1

Which is:
(2+3)x2 + (6−2)x + (5−1)

Add the like terms:
5×2 + 4x + 4

Other Examples:

(adding in columns)
Adding Several Polynomials – Using columns helps us to match the correct terms together in a complicated sum.

Example: Add (2×2 + 6y + 3xy) , (3×2 – 5xy – x) and (6xy + 5)

Line them up in columns and add:
2×2 + 6y + 3xy
3×2 – 5xy – x
6xy + 5
5×2 + 6y + 4xy – x + 5

Subtracting Polynomials

To subtract Polynomials, first reverse the sign of each term we are subtracting (in other words turn “+” into “-” and “-” into “+”), then add as usual.

Note: After subtracting 2xy from 2xy we ended up with 0, so there is no need to mention the “xy” term anymore.

Topic C: Multiplying Polynomials

To be able to multiply polynomials,
First, multiply each term in one polynomial by each term in the other polynomial using the distributive law.
Add the powers of the same variables using the exponent rule.
Then, simplify the resulting polynomial by adding or subtracting the like terms.

example of a polynomial
this one has 3 terms

1 term × 1 term (monomial times monomial)
To multiply one term by another term, first multiply the constants, then multiply each variable together and combine the result, like this (press play):

(Note: I used “·” to mean multiply. In Algebra we don’t like to use “×” because it looks too much like the letter “x”)

1 term × 2 terms (monomial times binomial)
Multiply the single term by each of the two terms, like this:

2 term × 1 terms (binomial times monomial)
Multiply each of the two terms by the single term, like this:

(I did that one a bit faster by multiplying in my head before writing it down)

2 terms × 2 terms (binomial times binomial)
We can multiply them in any order so long as each of the first two terms gets multiplied by each of the second two terms. But there is a handy way to help us remember to multiply each term called “FOIL”.

It stands for “Firsts, Outers, Inners, Lasts”:

Firsts: ac
Outers: ad
Inners: bc
Lasts: bd

So you multiply the “Firsts” (the first terms of both polynomials), then the “Outers”, etc. Let us try this on a more complicated example:

2 terms × 3 terms (binomial times trinomial)
“FOIL” won’t work here, because there are more terms now. But just remember: Multiply each term in the first polynomial by each term in the second polynomial

Like Terms

And always remember to add Like Terms:

Example: (x + 2y)(3x − 4y + 5)
(x + 2y)(3x − 4y + 5)
= 3×2 − 4xy + 5x + 6xy − 8y2 + 10y
= 3×2 + 2xy + 5x − 8y2 + 10y

Note: −4xy and 6xy are added because they are Like Terms.

Also note: 6yx means the same thing as 6xy

Long Method

Choose one polynomial (the longest is a good choice) and then:

• multiply it by the first term of the other polynomial, writing the result down
• then multiply it by the second term of the other polynomial, writing the result under the matching terms from the first multiplication
• then multiply it by the third term of the other polynomial (if any), etc.
• lastly, add up the columns.

By lining up the columns and being careful to put the terms under the correct columns, the job becomes “automatic”, and we can easily look back to see if we got it right, too.

But what happens if a polynomial is missing, say, an x term or an x2 term? Just leave that column blank!

Here is a more complicated example with blank gaps:

More than One Variable
So far, we have been multiplying polynomials with only one variable (x), but how do we handle polynomials with two or more variables (such as x and y)? What are the column headings?
Just ignore the columns in the question, and write down the answers as they come, always checking to see if we could put an answer under a matching answer:

Topic D: Dividing Polynomials

Sometimes it is easy to divide a polynomial by splitting it at the “+” and “−” signs, like this (press play):

When the polynomial was split into two parts we still had to keep the “/3” under each one.

Then the highlighted parts were “reduced” (6/3 = 2 and 3/3 = 1) to leave the answer of 2x-1

Here is another, slightly more complicated, example:

What happened?

• The 1st Term had x2 above and x below, which together becomes just x
• The 2nd Term had x above and below, so they canceled each other out
• We couldn’t simplify “1/3x” any further.

That is as far as we can get. But the answer is still “simpler”

Note: the result is a valid answer but is not a polynomial, because the last term (1/3x) has division by a variable (x).

Now, sometimes it helps to rearrange the top polynomial before dividing, as in this example:

Long Division

Dividing the Numerator by the Denominator
We can give each polynomial a name:

• the top polynomial is the numerator
• the bottom polynomial is the denominator

If you have trouble remembering, think denominator is down-nominator.

Write it down neatly:

• the denominator goes first,
• then a “)”,
• then the numerator with a line above

Both polynomials should have the “higher order” terms first (those with the largest exponents, like the “2” in x2).

Then:

• Divide the first term of the numerator by the first term of the denominator and put that in the answer.
• Multiply the denominator by that answer, put that below the numerator
• Subtract to create a new polynomial
• Repeat, using the new polynomial It is easier to show with an example!

Example:

Write it down neatly like below, then solve it step-by-step:

Check the answer:
Multiply the answer by the bottom polynomial, and we should get the top polynomial:

Remainders

The previous example worked perfectly, but that is not always so! Try this one:

After dividing we were left with “2”, this is the “remainder”.

The remainder is what is left over after dividing.

But we still have an answer: put the remainder divided by the bottom polynomial as part of the answer, like this:

“Missing” Terms

There can be “missing terms” (example: there may be an x3, but no x2). In that case either leave gaps or include the missing terms with a coefficient of zero.

Example:

Write it down with “0” coefficients for the missing terms, then solve it normally:

More than One Variable

So far, we have been dividing polynomials with only one variable (x), but we can handle polynomials with two or more variables (such as x and y) using the same method.
Example:

Solution:

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References:

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