1. Polynomials The final unit!
All real numbers can be represented as polynomials. So, like numbers, we will be learning how to add, subtract, multiply, divide, factor, raise to powers, etc. with polynomials.
Now, a lot of the beginning is review and renaming. We have actually been working on polynomials all year long!

2. Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms“

3. Polynomials
A polynomial looks like this:

4. A number, a variable, or the product/quotient of numbers/variables.
Term
A number, a variable, or the product/quotient of numbers/variables.
Review vocabulary. The following are examples of terms
4  is a term
3x  is a term
14x6  is a term
It is good to remember that terms are separated by addition and subtracting.
We have been dealing with terms all year long. When we solve equations, we are working with a group of terms. When we distribute, we are multiplying one term over several terms.

5. Coefficient: can by any real number… including zero.
A term has 3 components:
Exponent:
Can only be 0,1,2, 3, …
5 𝑥 3
Coefficient: can by any real number… including zero.
Variable: Usually denoted as x

6. 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

7. Remember

8. These are more examples of polynomials
-6y2 - (7/9)x
3xyz + 3xy2z - 0.1xz - 200y + 0.5
512v5+ 99w5
5 (which is really 5x0)

9. A polynomial can have: Terms that can be combined by adding, subtracting or multiplying.
A Polynomial can NOT have a division by a variable in it.

10. These are not allowed as a term in a polynomial
𝟐 𝒙+𝟐 is not, because dividing by a variable is not allowed
𝟏 𝒙 is not either
3xy-2 is not, because : 𝑦 −2 = 1 𝑦 2
𝒙 is not, because the exponent is "½"

11. These are allowed as a term in a polynomial
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 (= etc)

12. Monomial, binomial, and trinomial

13. There is also quadrinomial (4 terms) and quintinomial (5 terms), but those names are not often used.
Can Have Lots and Lots of Terms
Polynomials can have as many terms as needed, but not an infinite number of terms.

14. 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: 3x x3 + x6
The highest degree is 6, so that goes first, then 3, 2 and then the constant last:
x6 + 4x3 + 3x2 - 7

15. General Term of a polynomial

16. Degree of a Term The exponent of the variable.
We will find them only for one-variable terms.
New vocabulary!
It is a simple concept. It is just the exponent of the variable. You can find the degree of multiply variable terms, but we don’t deal with them in algebra 1.
Now these degrees are very important. They tell you which family of functions a term belongs to and will have a big impact on polynomials.

17. Term Degree of Term 3 4x 1 -5x2 2 18x5 51 2 5
The only tricky thing about determining the degree of a term…
Terms without variables have a degree of zero. You can imagine that there was a variable there, but it was raised to the zero power… like 3x0 … because following the order of operations… x0 = 1 so 3x0 = 3.
Now, you also have to remember that a variable that doesn’t have an exponent still has a degree… this time of 1. Remember mathematicians don’t write coefficients or exponents of 1. Those are understood to be there and are like “Phantom Ones”

18. Polynomial
A term or the sum/difference of terms which contain only 1 variable. The variable cannot be in the denominator of a term.
The key vocabulary word of the day…
Things that are Polynomials  numbers by themselves, terms by themselves, sums of terms, difference of terms… these are all polynomials. They can have fractions, decimals, 1 term, many terms… 14
1.2x2
2x + 4
½ x4 – 4x3 + 2x2 + 7x - 100
5x2 + 3x – 7
Not Polynomials  no variables as the exponent, no negative exponents, no variables in denominators, no multiple variables. 2x
3x + x-3
4/x + 7
3xy + 5xy2

19. The degree of the term with the highest degree.
Degree of a Polynomial
The degree of the term with the highest degree.
Basically the highest exponent in a polynomial. This is very important because that highest degree has the biggest impact on the value of the expression. If we were to put a “y=“ in front of the polynomial to turn it into a function, that highest degreed term would tell you which family it belongs in, what the graph is going to look like, etc.

20. Degree of Polynomial Polynomial 2 6x2 - 3x + 2 4 15 - 4x + 5x4 1
Examples

21. Standard Form of a Polynomial
A polynomial written so that the degree of the terms decreases from left to right and no terms have the same degree.
Very much common sense. Basically Standard form is simplified (like terms are combined), you just have to make sure to write the terms in the correct order. The highest degree comes first… the rest follow in order of decreasing degree.

22. Not Standard 6x + 3x2 - 2 15 - 4x + 5x4 x + 10 + x 1 + x2 + x + x3
Examples

23. Naming Polynomials
Polynomials are named or classified by their degree and the number of terms they have.
Polynomials have 2 names. The first name is by family (degree). It is like the name you would give a type of function. The second name tells you how many terms there are in the polynomial.
When giving the full name of a polynomial, you have to include both the name by degree and the name by the number of terms

24. 7 5x + 2 4x2 + 3x - 4 6x3 - 18 1 2 3 constant linear quadratic cubic
Polynomial 7
5x + 2
4x2 + 3x - 4
6x3 - 18
Degree 1 2 3
Degree Name
constant
linear
quadratic
cubic
These are the first (family, by degree) names. These names match up with types of functions we have worked with. Now, the names go beyond the types listed here, but are rarely used. The box tells you how to deal with higher order (higher degreed) polynomials.
For degrees higher than 3 say:
“of degree n” or “nth degree”
x5 + 3x “of degree 5” – or – “5th degree”
x “of degree 8” – or – “8th degree”

25. 7 5x + 2 4x2 + 3x - 4 6x3 - 18 1 2 3 monomial binomial trinomial
Polynomial 7
5x + 2
4x2 + 3x - 4
6x3 - 18
# Terms 1 2 3
# Terms Name
monomial
binomial
trinomial
These are the names by the number of terms. “nomial” means number of terms. “mon” means 1… “bi” means two…”tri” means 3… there are more, but most people only use up to trinomial. Above that, well, look in the box. It tells you how to name something with more than 3 terms.
For more than 3 terms say:
“a polynomial with n terms” or “an n-term polynomial”
11x8 + x5 + x4 - 3x3 + 5x2 - 3
“a polynomial with 6 terms” – or – “a 6-term polynomial”

26. Polynomial Name -14x3 -1.2x2 -1 7x - 2 3x3+ 2x - 8 2x2 - 4x + 8 x4 + 3
cubic monomial
quadratic monomial
constant monomial
linear binomial
cubic trinomial
quadratic trinomial
4th degree binomial
Here are examples of the full names of polynomials.
Remember we use the family (degree) name first. It is the most important name.

27. Adding and Subtracting Polynomials
To add or subtract polynomials, simply combine like terms.
So, nothing really new here. We’ve been doing this for a long while. I’m just going to add more terms than you are used to and terms of higher degree.

28. (5x2 - 3x + 7) + (2x2 + 5x - 7)
= 7x2 + 2x
(3x3 + 6x - 8) + (4x2 + 2x - 5)
= 3x3 + 4x2 + 8x - 13
Example… this is called the horizontal method…where you keep the problem written horizontally and mentally group the terms together by degree. You can also write the rearrangement down (by commuting the terms).
Just be sure to combine the terms with the correct terms. For example, the second example, it is common for student to accidentally combine the cubic term and the quadratic term.

29. (2x3 + 4x2 - 6) – (3x3 + 2x - 2) (2x3 + 4x2 - 6) + (-3x3 + -2x - -2)
More examples… this time with subtraction. You have to remember that every term of that second polynomial is being subtracted. So it is useful to change the problem to an addition one before moving on. You basically have to change every sign in the polynomial that you are subtracting. In essence, you are distributing a negative one.
(7x3 - 3x + 1) + (-x3 - -4x2 - -2)
= 6x3 + 4x2 - 3x + 3

30. TRY IT THE VERTICAL WAY! 7y2 – 3y + 4 + 8y2 + 3y – 4
2x3 – 5x2 + 3x – 1
– (8x3 – 8x2 + 4x + 3)
15y2 + 0y = 15y2
TRY IT THE VERTICAL WAY!
Now, some people like to arrange their addition and subtracting in the “old school” way… which is how you were taught to add and subtract in elementary school. You just write the problem so that the like terms line up with each other vertically… then add or subtract as necessary. I put the subtraction in parenthesis to remind myself that every term needs to be subtracted. Sometimes if you don’t , you just think that the coefficient of the first term is negative.
–6x3 + 3x2 – x – 4

31. (7y3 +2y2 + 5y – 1) + (5y3 + 7y) 7y3 + 2y2 + 5y – 1
More examples… the zeroes are placed only as place holders. They are unnecessary

32. Rewrite in standard form!
(b4 – 6 + 5b + 1) + (8b4 + 2b – 3b2)
Rewrite in standard form!
b4 + 0b3 + 0b2 + 5b – 5
+ 8b4 + 0b3 – 3b2 + 2b + 0
Even more examples
9b4 + 0b3 – 3b2 + 7b – 5
= 9b4 – 3b2 + 7b – 5

33. Polynomials: Algebra Tiles
Algebra tiles are tools that help one represent a polynomial.
Ah.. Algebra tiles. I am obligated by the state of Texas to show you algebra tiles. They are used to represent polynomials of degrees no higher than 2.
The red tiles are all negatives…. –x2 , -x, -1 (in order from left to right)
The blue, green, and yellow are the positives… x2 , x, 1 (in order from left to right)
You can use the tiles to show a picture of a polynomial, to show how addition/subtractions works… but what they are really good for is showing how multiplication works… Let’s start with the basics.

34. Polynomials: Algebra Tiles
What polynomial is represented below?
So, what is the polynomial represented above?
2x2 + 2x – 4x + 8 – 3
Simplified (because red/green and red/yellow cancel)
2x2 – 2x + 5

35. Polynomials: Multiplying by a Monomial w/tiles
When you multiply, you are really finding the areas of rectangles. The length and width are the factors and the product is the area.
For example multiply 2x (3x + 1)
3x+1
Now the real reason why we use these tiles…
They are really good at having you understand how to multiply polynomials. Each factor is a dimension of a rectangle. So 2x is the width and 3x + 1 is the length.
Then you add the pieces to make a rectangle… x * x is x2 so you put little x2 pieces wherever you see an x *x. When you see an x*1 you get x, so you put a little green tile there. You end up with six x2 and two x, which is the product we knew it would be to begin with (because we know how to distribute)
This seems annoying now, but will help you out when I ask you to multiply two binomials.
6x2 + 2x 2x

36. Multiplying Polynomials
Distribute and FOIL

37. Polynomials * Polynomials
Multiplying a Polynomial by another Polynomial requires more than one distributing step.
Multiply: (2a + 7b)(3a + 5b)
Distribute 2a(3a + 5b) and distribute 7b(3a + 5b):
6a2 + 10ab
21ab + 35b2
Then add those products, adding like terms:
6a2 + 10ab + 21ab + 35b2
= 6a2 + 31ab + 35b2

38. Polynomials * Polynomials
An alternative is to stack the polynomials and do long multiplication.
(2a + 7b)
x (3a + 5b)
(2a + 7b)(3a + 5b)
(2a + 7b)
x (3a + 5b)
Multiply by 5b, then by 3a:
When multiplying by 3a, line up the first term under 3a.
21ab + 35b2 +
6a2 + 10ab
Add like terms:
6a2 + 31ab + 35b2

39. Polynomials * Polynomials
Multiply the following polynomials:

40. Polynomials * Polynomials
(x + 5)
x (2x + -1)
-x + -5
2x2 + 10x +
2x2 + 9x + -5
(3w + -2)
x (2w + -5)
-15w + 10 +
6w2 + -4w
6w w + 10

41. Polynomials * Polynomials
(2a2 + a + -1)
x (2a2 + 1)
2a2 + a + -1 +
4a4 + 2a3 + -2a2
4a4 + 2a3 + a + -1

42. Types of Polynomials
We have names to classify polynomials based on how many terms they have:
Monomial: a polynomial with one term
Binomial: a polynomial with two terms
Trinomial: a polynomial with three terms

43. (2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 = 8x2 + -2x + -15
F.O.I.L.
There is an acronym to help us remember how to multiply two binomials without stacking them.
(2x + -3)(4x + 5)
F : Multiply the First term in each binomial. 2x • 4x = 8x2
O : Multiply the Outer terms in the binomials. 2x • 5 = 10x
I : Multiply the Inner terms in the binomials. -3 • 4x = -12x
L : Multiply the Last term in each binomial. -3 • 5 = -15
(2x + -3)(4x + 5) = 8x2 + 10x + -12x = 8x2 + -2x + -15

44. Use the FOIL method to multiply these binomials:
1) (3a + 4)(2a + 1)
2) (x + 4)(x - 5)
3) (x + 5)(x - 5)
4) (c - 3)(2c - 5)
5) (2w + 3)(2w - 3)

45. Use the FOIL method to multiply these binomials:
1) (3a + 4)(2a + 1) = 6a2 + 3a + 8a + 4 = 6a2 + 11a + 4
2) (x + 4)(x - 5) = x2 + -5x + 4x = x2 + -1x + -20
3) (x + 5)(x - 5) = x2 + -5x + 5x = x
4) (c - 3)(2c - 5) = 2c2 + -5c + -6c + 15 = 2c c + 15
5) (2w + 3)(2w - 3) = 4w2 + -6w + 6w + -9 = 4w2 + -9