1. Operations on Polynomials

2. Definition
A monomial is the product of real numbers and one or more variables with whole number exponents
Some examples
2 𝑥 2
1 3 𝑎 𝑏 3 3
𝑤 𝑥 2 𝑦 3 𝑧 4
Although a monomial can have more than one variable, as two of the above examples show, we will mostly be concerned with monomials including a single variable (usually x)

3. Definition A polynomial is the sum of one or more monomials
Note that a monomial by itself is also a polynomial (one with a special name); monomials are the same as the terms of a polynomial
Two other polynomials with special names are binomial (2 terms) and trinomial (3 terms)
Other examples
3 𝑎 2 𝑏−4𝑐 𝑑 2 (a binomial)
4 𝑥 3 +2 𝑥 2 +𝑥−1
6 (a monomial)
𝑥+𝑦+𝑧 (a trinomial)

4. Polynomials
We will mostly be concerned with polynomials of one variable (usually x)
So for our purposes, a polynomial expression is one of the form 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +⋯+ 𝑎 1 𝑥+ 𝑎 0
Here, each 𝑎 𝑖 (where i is a number between 0 and n inclusive) is a real number and called the coefficient, and n is a whole number
The values 𝑎 𝑛 is called the lead coefficient and must not equal zero
The value 𝑎 0 is the constant; note that it is the only monomial without a variable x

5. Polynomials
Let’s look at an example to see how this definition works 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +⋯+ 𝑎 1 𝑥+ 𝑎 0
Suppose that we have a polynomial where 𝑛=4 is the largest exponent on the variable x
Our polynomial is then 𝑎 4 𝑥 4 + 𝑎 3 𝑥 3 + 𝑎 2 𝑥 2 + 𝑎 1 𝑥+ 𝑎 0
Here, 𝑎 4 is the lead coefficient and 𝑎 0 is the constant
In this example, no particular values have been given to the coefficients 𝑎 4 , 𝑎 3 , 𝑎 2 , 𝑎 1 , nor to 𝑎 0

6. Polynomials
You will be working with polynomials where the coefficients and the constant have values, usually integer values
In the previous example, let’s assign the following values 𝑎 4 =2, 𝑎 3 =1, 𝑎 2 =0, 𝑎 1 =−3, 𝑎 0 =−5
Then the polynomial becomes 2 𝑥 4 + 𝑥 3 −3𝑥−5
Note that the term with 𝑥 2 is missing since its coefficient is zero
The lead coefficient is 2 while the constant is −5

7. Polynomials
Because of the commutative property, 2 𝑥 4 + 𝑥 3 −3𝑥−5=−5+2 𝑥 4 −3𝑥+ 𝑥 3
However, most of the time we will want to write polynomials as shown on the left side of the equal sign
That is, the term with the largest exponent is written first, then the term with the next largest exponent is written second, and so on, the last term being the constant
This is know as the standard form of a polynomial

8. Guided Practice
Use the commutative property of addition to rewrite the following polynomials in standard form
−4 𝑥 2 +2 𝑥 4 −3 𝑥 5 −1
−7+2 𝑥 2
3 𝑥 2 −𝑥+4 𝑥 3
3+2 𝑥 4 −4𝑥

9. Guided Practice
Use the commutative property of addition to rewrite the following polynomials in standard form
−4 𝑥 2 +2 𝑥 4 −3 𝑥 5 −1=−3 𝑥 5 +2 𝑥 4 −4 𝑥 2 −1
−7+2 𝑥 2 =2 𝑥 2 −7
3 𝑥 2 −𝑥+4 𝑥 3 =4 𝑥 3 +3 𝑥 2 −𝑥
3+2 𝑥 4 −4𝑥=2 𝑥 4 −4𝑥+3

10. Degree of a Polynomial
A number that will be useful in classifying polynomials later is the degree of a polynomial
The degree of a polynomial is the value of its greatest exponent
We define the degree of a constant to be zero, while zero itself has no degree

11. Degree of a Polynomial The following polynomials are of degree 4
𝑥 4
− 𝑥 4 −3 𝑥 3 +4 𝑥 2 +7𝑥+1
4−6 𝑥 4
2 𝑥 4 +𝑥
We could also say that the polynomials listed above are 4th degree polynomials

12. Guided Practice Give the degree of the polynomial. 3𝑥−5 𝑥 3 +7+ 𝑥 6 5
𝑥−1
4𝑥−1+2 𝑥 2

13. Guided Practice Give the degree of the polynomial.
(Note that it is easier to tell at a glance the degree of a polynomial if it is written in standard form)

14. Operations on Polynomials
Note the similarity between a polynomial, say 3 𝑥 4 +2 𝑥 2 +5𝑥+1, and the numeral 30251
They are not the same (necessarily), but if we expand the numeral we get 3⋅ ⋅ ⋅10+1
This suggest that we should think of a polynomial as, in a sense, a single number, and we can perform operations on pairs of polynomials
Specifically, we can add, subtract, multiply, and divide them

15. Adding/Subtracting Polynomials
When adding numbers in columns, as shown below, why is it necessary to line the digits up in a particular way?
4235
+41
The result will be incorrect if, say, the 4 in 41 is lined up with the 4 in 4235, and the 1 lined up with the 2, but why?
Expanding both numbers gives us a clue
4⋅ ⋅ ⋅ ⋅10+1

16. Adding/Subtracting Polynomials
Following this example, how would you expect to add the following
4 𝑥 3 +2 𝑥 2 +3𝑥+5 +(4𝑥+1)

17. Adding/Subtracting Polynomials
Following this example, how would you expect to add the following
4 𝑥 3 +2 𝑥 2 +3𝑥+5 +(4𝑥+1)
We can take a lesson from how we numerals and add these in columns
4 𝑥 3 +2 𝑥 2 +3𝑥 𝑥+1
The result is
4 𝑥 3 +2 𝑥 2 +7𝑥+6

18. Adding/Subtracting Polynomials
Although our use of analogy to the addition algorithm you learned in elementary school suggests how to proceed with addition and subtraction of polynomials, it is not enough
In fact, we must turn to the distributive property to come up with a rule for adding/subtracting polynomials
Is the following true:
5+3 𝑥=5𝑥+3𝑥
Is this true:
5+3 𝑥=8𝑥

19. Adding/Subtracting Polynomials
If 5+3 𝑥=5𝑥+3𝑥 and 5+3 𝑥=8𝑥, then is this true?
5𝑥+3𝑥=8𝑥
We can formulate a rule: we will say that all terms with the same variable and the same exponent on the variable are like terms
Now we can say simply that like terms can be simplified by adding/subtracting their coefficients
We are now ready to apply this to addition/subtraction of polynomials

20. Adding/Subtracting Polynomials
Example: simplify the following
3 𝑥 3 −2 𝑥 2 +10𝑥−5 + 6 𝑥 2 −7𝑥+1
One way to do this is to add in columns, lining up the like terms
3 𝑥 3 −2 𝑥 2 +10𝑥− 𝑥 2 −7𝑥+1
Now combine the coefficients; this is like adding numerals in columns, but remember that there is no carrying
The result is 3 𝑥 3 +4 𝑥 2 +3𝑥−4

21. Adding/Subtracting Polynomials
If a polynomial is “missing” a term, then use zero in its place
Example: add 4 𝑥 4 +2 𝑥 2 −1 + − 𝑥 4 +3 𝑥 3 +2𝑥+1
4 𝑥 4 +0 𝑥 3 +2 𝑥 2 +0𝑥−1 − 𝑥 4 +3 𝑥 3 +0 𝑥 2 +2𝑥+1
3 𝑥 4 +3 𝑥 3 +2 𝑥 2 +2𝑥+0
We can leave zero out, so the final sum is
3 𝑥 4 +3 𝑥 3 +2 𝑥 2 +2𝑥

22. Adding/Subtracting Polynomials
Subtracting polynomials requires special attention
Remember that −𝑛=−1⋅𝑛 and also that 𝑎−𝑏=𝑎+(−𝑏)
Now, consider this difference of polynomials
5 𝑥 2 +3𝑥−4 −(−2 𝑥 3 +4𝑥−1)
Using our definition of subtraction this becomes
5 𝑥 2 +3𝑥−4 + − −2 𝑥 3 +4𝑥−1
But since the opposite of a number is the same as −1 times the number
5 𝑥 2 +3𝑥−4 + −1 −2 𝑥 3 +4𝑥−1

23. Adding/Subtracting Polynomials
5 𝑥 2 +3𝑥−4 + −1 −2 𝑥 3 +4𝑥−1
Now using the distributive property
5 𝑥 2 +3𝑥−4 + 2 𝑥 3 −4𝑥+1
We can now use column addition
2 𝑥 3 +0 𝑥 2 −4𝑥+1
𝑥 2 +3𝑥−4
2 𝑥 3 +5 𝑥 2 −𝑥−3

24. Adding/Subtracting Polynomials
The effect of the previous example was to change the signs inside the parentheses and change subtraction to addition
5 𝑥 2 +3𝑥−4 − −2 𝑥 3 +4𝑥−1
This is the same as
5 𝑥 2 +3𝑥−4 +(2 𝑥 3 −4𝑥+1)

25. Guided Practice Add or subtract as required.
3 𝑥 3 −2 𝑥 −5 𝑥 3 +6𝑥+3
7𝑥−1 − 2 𝑥 2 +4𝑥+7
𝑥 4 +4 𝑥 2 −5 − 𝑥 4 +3 𝑥 3 −4 𝑥 2 −5
3 𝑥 4 −1 −(5 𝑥 4 +2 𝑥 3 +3𝑥−3)

26. Guided Practice Add or subtract as required.
3 𝑥 3 −2 𝑥 −5 𝑥 3 +6𝑥+3 =−2 𝑥 3 −2 𝑥 2 +6𝑥+7
7𝑥−1 − 2 𝑥 2 +4𝑥+7 =−2 𝑥 2 +3𝑥−8
𝑥 4 +4 𝑥 2 −5 − 𝑥 4 +3 𝑥 3 −4 𝑥 2 −5 =−3 𝑥 3 +8 𝑥 2
3 𝑥 4 −1 − 5 𝑥 4 +2 𝑥 3 +3𝑥−3 =−2 𝑥 4 −2 𝑥 3 −3𝑥+2

27. Practice 2
Handout