The document discusses polynomials and factoring polynomials. It defines polynomials as expressions with terms added or subtracted, where terms are products of numbers and variables with exponents. It provides examples of monomials, binomials, trinomials, and polynomials based on the number of terms. It also discusses finding the greatest common factor of a polynomial to factor out a monomial.

1. Polynomials andPolynomials and
FactoringFactoring
The basic building blocks ofThe basic building blocks of
algebraic expressionsalgebraic expressions

2. The height in feet of
a fireworks launched
straight up into the air
from (s) feet off the
ground at velocity (v) after
(t) seconds is given by the
equation:
-16t2
+ vt + s
Find the height of a
firework launched from a 10
ft platform at 200 ft/s
after 5 seconds.
-16t2
+ vt + s
-16(5)2
+ 200(5) + 10
=400 + 1600 + 10
610 feet

3. In regular math books, this is called
“substituting” or “evaluating”… We are given
the algebraic expression below and asked to
evaluate it.
x2
– 4x + 1
We need to find what this equals when we put a
number in for x.. Like
x = 3
Everywhere you see an x… stick in a 3!
x2
– 4x + 1
= (3)2
– 4(3) + 1
= 9 – 12 + 1
= -2

4. You try a couple
Use the same expression but let
x = 2 and
x = -1
What about x = -5?
Be careful with the negative! Use ( )!
x2
– 4x + 1
= (-5)2
– 4(-5) + 1
= 46

5. That critter in the last slide is a polynomial.
x2
– 4x + 1
Here are some others
x2
+ 7x – 3
4a3
+ 7a2
+ a
nm2
– m
3x – 2
5

6. For now (and, probably, forever) you
can just think of a polynomial as a
bunch to terms being added or
subtracted. The terms are just
products of numbers and letters
with exponents. As you’ll see later
on, polynomials have cool graphs.

7. Some math words to know!
monomial – is an expression that is a number, a
variable, or a product of a number and one or
more variables. Consequently, a monomial has no
variable in its denominator. It has one term.
(mono implies one).
13, 3x, -57, x2
, 4y2
, -2xy, or 520x2
y2
(notice: no negative exponents, no fractional
exponents)
binomial – is the sum of two monomials. It has two
unlike terms (bi implies two).
3x + 1, x2
– 4x, 2x + y, or y – y2

8. trinomial – is the sum of three monomials. It has
three unlike terms. (tri implies three).
x2
+ 2x + 1, 3x2
– 4x + 10, 2x + 3y + 2
polynomial – is a monomial or the sum (+) or
difference (-) of one or more terms.
(poly implies many).
x2
+ 2x, 3x3
+ x2
+ 5x + 6, 4x + 6y + 8
• Polynomials are in simplest form when they contain no like
terms. x2
+ 2x + 1 + 3x2
– 4x when simplified
becomes 4x2
– 2x + 1
• Polynomials are generally written in descending order.
Descending: 4x2
– 2x + 1 (exponents of variables decrease
from left to right)
The ending of these
words “nomial” is Greek
for “part”.
Constants like 12 are monomials
since they can be written as 12x0
=
12 · 1 = 12 where the variable is x0
.

9. The degree of a monomial - is the sum of the
exponents of its variables. For a nonzero
constant, the degree is 0. Zero has no degree.
Find the degree of each monomial
a) ¾x degree: 1 ¾x = ¾x1
. The exponent is 1.
b) 7x2
y3
degree: 5 The exponents are 2 and 3. Their sum is 5.
c) -4 degree: 0 The degree of a nonzero constant is 0.

10. Here’s a polynomial
2x3
– 5x2
+ x + 9
Each one of the little product things is a “term”.
2x3
– 5x2
+ x + 9
So, this guy has 4 terms.
2x3
– 5x2
+ x + 9
The coefficients are the numbers in front of the letters.
2x3
- 5x2
+ x + 9
term term term term
2 5 1 9
We just pretend
this last guy has a
letter behind him.
Remember
x = 1 · x
NEXT

11. Since “poly” means many, when there is only one term,
it’s a monomial:
5x
When there are two terms, it’s a binomial:
2x + 3
When there are three terms, it a trinomial:
x2
– x – 6
So, what about four terms? Quadnomial? Naw, we
won’t go there, too hard to pronounce.
This guy is just called a polynomial:
7x3
+ 5x2
– 2x + 4 NEXT

12. So, there’s one word to remember to classify:
degree
Here’s how you find the degree of a polynomial:
Look at each term,
whoever has the most letters wins!
3x2
– 8x4
+ x5
This is a 7th
degree polynomial:
6mn2
+ m3
n4
+ 8
This guy has 5
letters…
The degree is 5.
This guy has 7 letters…
The degree is 7 NEXT

13. This is a 1st
degree polynomial
3x – 2
What about this dude?
8
How many letters does he have? ZERO!
So, he’s a zero degree polynomial
This guy has 1
letter…
The degree is 1.
This guy has no
letters…
The degree is 0.
By the way, the
coefficients don’t
have anything to
do with the
degree.
Before we go, I want you to know that
Algebra isn’t going to be just a bunch of weird
words that you don’t understand. I just
needed to start with some vocabulary so you’d
know what the heck we’re talking about!

14. 3x4
+ 5x2
– 7x + 1
The polynomial above is in standard form.
Standard form of a polynomial - means that
the degrees of its monomial terms decrease
from left to right.
term
termtermterm
Polynomial Degree Name using
Degree
Number of
Terms
Name using
number of
terms
7x + 4 1 Linear 2 Binomial
3x2
+ 2x + 1 2 Quadratic 3 Trinomial
4x3
3 Cubic 1 Monomial
9x4
+ 11x 4 Fourth degree 2 Binomial
5 0 Constant 1 monomial
Once you simplify a polynomial by
combining like terms, you can name the
polynomial based on degree or number of
monomials it contains.

15. Classifying Polynomials
Write each polynomial in standard form. Then name each
polynomial based on its degree and the number of terms.
a) 5 – 2x
-2x + 5 Place terms in order.
linear binomial
b) 3x4
– 4 + 2x2
+ 5x4
Place terms in order.
3x4
+ 5x4
+ 2x2
– 4 Combine like terms.
8x4
+ 2x2
– 4
4th
degree trinomial

16. Write each polynomial in standard
form. Then name each polynomial
based on its degree and the
number of terms.
a) 6x2
+ 7 – 9x4
b) 3y – 4 – y3
c) 8 + 7v – 11v

17. Adding and SubtractingAdding and Subtracting
PolynomialsPolynomials
The sum or differenceThe sum or difference

18. Just as you can perform operations on
integers, you can perform operations on
polynomials. You can add polynomials using two
methods. Which one will you choose?
Closure of polynomials under addition or subtraction
The sum of two polynomials is a polynomial.
The difference of two polynomials is a polynomial.

19. Addition of
Polynomials
Method 1 (vertically)
Line up like terms. Then add the coefficients.
4x2
+ 6x + 7 -2x3
+ 2x2
– 5x + 3
2x2
– 9x + 1 0 + 5x2
+ 4x - 5
6x2
– 3x + 8 -2x3
+ 7x2
– x - 2
Method 2 (horizontally)
Group like terms. Then add the coefficients.
(4x2
+ 6x + 7) + (2x2
– 9x + 1) = (4x2
+ 2x2
) + (6x – 9x) + (7 + 1)
= 6x2
– 3x + 8
Example 2:
(-2x3
+ 0) + (2x2
+ 5x2
) + (-5x + 4x) + (3 – 5)
You can rewrite each polynomial,
inserting a zero placeholder for
the “missing” term.
Example 2
Use a zero placeholder

20. Simplify each sum
• (12m2
+ 4) + (8m2
+ 5)
• (t2
– 6) + (3t2
+ 11)
• (9w3
+ 8w2
) + (7w3
+ 4)
• (2p3
+ 6p2
+ 10p) + (9p3
+ 11p2
+ 3p )
Remember
Use a zero as a placeholder
for the “missing” term.
Word Problem

21. Find the perimeter of
each figure
9c - 10
5c + 2
17x - 6
5x
+
1
9x
8x-2
Recall that the
perimeter of a figure is
the sum of all the sides.

22. Subtracting
Polynomials
Earlier you learned that subtraction means to add the
opposite. So when you subtract a polynomial, change
the signs of each of the terms to its opposite. Then
add the coefficients.
Method 1 (vertically)
Line up like terms. Change the signs of the second polynomial, then
add. Simplify (2x3
+ 5x2
– 3x) – (x3
– 8x2
+ 11)
2x3
+ 5x2
– 3x 2x3
+ 5x2
– 3x
-(x3
– 8x2
+ 0 + 11) -x3
+ 8x2
+ 0 - 11
x3
+13x2
– 3x - 11
Remember,
subtraction is adding
the opposite.
Method 2

23. Method 2 (horizontally)
Simplify (2x3
+ 5x2
– 3x) – (x3
– 8x2
+ 11)
Write the opposite of each term.
2x3
+ 5x2
– 3x – x3
+ 8x2
– 11
Group like terms.
(2x3
– x3
) + (5x2
+ 8x2
) + (3x + 0) + (-11 + 0) =
x3
+ 13x2
+ 3x - 11 =
x3
+ 13x2
+ 3x - 11

24. Simplify each
subtraction
• (17n4
+ 2n3
) – (10n4
+ n3
)
• (24x5
+ 12x) – (9x5
+ 11x)
• 6c – 5 2b + 6 7h2
+ 4h - 8
-(4c + 9) -(b + 5) -(3h2
– 2h + 10)

25. Multiplying and FactoringMultiplying and Factoring
Using the Distributive PropertyUsing the Distributive Property

26. Observe the rectangle below. Remember that
the area A of a rectangle with length l and
width w is A = lw. So the area of this
rectangle is (4x)(2x), as shown.
****************************
The rectangle above shows the example that
4x = x + x + x + x and 2x = x + x
4x
2x
A = lw
A = (4x)(2x)
x + x + x + x
x
+
x
NEXT

27. We can further divide the rectangle into
squares with side lengths of x.
x + x + x + x
x
+
x
x2
x2
x2
x2
x2
x2
x2
x2
x + x + x + x
x
+
x
Since each side of the squares
are x, then x · x = x2
By applying the area formula
for a rectangle, the area of the
rectangle must be (4x)(2x).
This geometric model suggests the following
algebraic method for simplifying the product
of (4x)(2x).
(4x)(2x) = (4 · x)(2 · x) = (4 · 2)(x · x) = 8x2
NEXT
Commutative Property Associative Property

28. To simplify a product of monomials
(4x)(2x)
• Use the Commutative and Associative Properties
of Multiplication to group the numerical
coefficients and to group like variable;
• Calculate the product of the numerical
coefficients; and
• Use the properties of exponents to simplify the
variable product.
Therefore (4x)(2x) = 8x2
(4x)(2x) = (4 · 2)(x · x ) =
(4 · 2) = 8
(x · x) = x1
· x1
= x1+1
= x2

29. You can also use the Distributive Property for
multiplying powers with the same base when
multiplying a polynomial by a monomial.
Simplify -4y2
(5y4
– 3y2
+ 2)
-4y2
(5y4
– 3y2
+ 2) =
-4y2
(5y4
) – 4y2
(-3y2
) – 4y2
(2) = Use the Distributive Property
-20y2 + 4
+ 12y2 + 2
– 8y2
= Multiply the coefficients and add the
-20y6
+ 12y4
– 8y2
exponents of powers with the same base.
Remember,
Multiply powers with the same base:
35
· 34
= 35 + 4
= 39

30. Simplify each product.
a) 4b(5b2
+ b + 6)
b) -7h(3h2
– 8h – 1)
c) 2x(x2
– 6x + 5)
d) 4y2
(9y3
+ 8y2
– 11)
Remember,
Multiplying powers with the same base.

31. Factoring a Monomial
from a Polynomial Factoring a polynomial
reverses the
multiplication process.
To factor a monomial
from a polynomial, first
find the greatest
common factor (GCF) of
its terms.
Find the GCF of the terms of:
4x3
+ 12x2
– 8x
List the prime factors of each term.
4x3
= 2 · 2 · x · x x
12x2
= 2 · 2 · 3 · x · x
8x = 2 · 2 · 2 · x
The GCF is 2 · 2 · x or 4x.

32. Find the GCF of the terms of each polynomial.
a) 5v5
+ 10v3
b) 3t2
– 18
c) 4b3
– 2b2
– 6b
d) 2x4
+ 10x2
– 6x

33. Factoring Out a
Monomial
Factor 3x3
– 12x2
+ 15x
Step 1
Find the GCF
3x3
= 3 · x · x · x
12x2
= 2 · 2 · 3 · x · x
15x = 3 · 5 · x
The GCF is 3 · x or 3x
Step 2
Factor out the GCF
3x3
– 12x2
+ 15x
= 3x(x2
) + 3x(-4x) + 3x(5)
= 3x(x2
– 4x + 5)
To factor a polynomial
completely, you must factor
until there are no common
factors other than 1.

34. Use the GCF to factor each polynomial.
a) 8x2
– 12x
b) 5d3
+ 10d
c) 6m3
– 12m2
– 24m
d) 4x3
– 8x2
+ 12x
Try to factor mentally by
scanning the coefficients of
each term to find the GCF.
Next, scan for the least power
of the variable.

35. Multiplying BinomialsMultiplying Binomials
Using the infamous FOIL methodUsing the infamous FOIL method

36. Using the
Distributive
Property
As with the other
examples we have
seen, we can also
use the Distributive
Property to find the
product of two
binomials.
Simplify: (2x + 3)(x + 4)
(2x + 3)(x + 4) =
2x(x + 4) + 3(x + 4) =
2x2
+ 8x + 3x + 12 =
2x2
+ 11x + 12
Now Distribute 2x and 3
Distribute x + 4

37. Simplify each
product.
a) (6h – 7)(2h + 3)
b) (5m + 2)(8m – 1)
c) (9a – 8)(7a + 4)
d) (2y – 3)(y + 2)

38. Multiplying using FOIL
Another way to organize multiplying two binomials is
to use FOIL, which stands for,
“First, Outer, Inner, Last”. The term FOIL is a
memory device for applying the Distributive
Property to the product of two binomials.
Simplify (3x – 5)(2x + 7)
First Outer Inner Last
= (3x)(2x) + (3x)(7) – (5)(2x) – (5)(7)
(3x – 5)(2x + 7) = 6x2
+ 21x - 10x - 35
= 6x2
+ 11x - 35
The product is 6x2
+ 11x - 35

39. Simplify each product
using FOIL
a) (3x + 4)(2x + 5)
b) (3x – 4)(2x + 5)
c) (3x + 4)(2x – 5)
d) (3x – 4)(2x – 5)
Remember,
First, Outer, Inner,
Last

40. Applying
Multiplication of
Polynomials.
Find the area of the
shaded (beige) region.
Simplify.
area of outer rectangle =
(2x + 5)(3x + 1)
area of orange rectangle =
x(x + 2)
area of shaded region
= area of outer rectangle – area of
orange portion
(2x + 5)(3x + 1) – x(x + 2) =
6x2
+ 15x + 2x + 5 – x2
– 2x =
6x2
– x2
+ 15x + 2x – 2x + 5 =
5x2
+ 17x + 5
2x + 5
x + 2
x
3x+1
Use the FOIL method to
simplify (2x + 5)(3x + 1)
Use the Distributive Property
to simplify –x(x + 2)

41. Find the area of the
shaded region.
Simplify.
Find the area of the green shaded region. Simplify.
5x + 8
6x+2
5x
x + 6

42. FOIL works when you are multiplying two binomials.
However, it does not work when multiplying a trinomial
and a binomial.
(You can use the vertical or horizontal method to distribute each term.)
Simplify (4x2
+ x – 6)(2x – 3)
Method 1 (vertical)
4x2
+ x - 6
2x - 3
-12x2
- 3x + 18 Multiply by -3
8x3
+ 2x2
- 12x Multiply by 2x
8x3
- 10x2
- 15x + 18 Add like terms
Remember multiplying
whole numbers.
312
x 23
936
624
7176

43. Multiply using the
horizontal method.
(2x – 3)(4x2
+ x – 6)
= 2x(4x2
) + 2x(x) + 2x(-6) – 3(4x2) – 3(x) – 3(-6)
= 8x3
+ 2x2
– 12x – 12x2
– 3x + 18
= 8x3
-10x2
- 15x + 18
The product is 8x3
– 10x2
– 15x + 18
Drawing arrows
between terms can
help you identify all six
products.
Method 2 (horizontal)

44. Simplify using the Distributive Property.
a) (x + 2)(x + 5)
b) (2y + 5)(y – 3)
c) (h + 3)(h + 4)
Simplify using FOIL.
a) (r + 6)(r – 4)
b) (y + 4)(5y – 8)
c) (x – 7)(x + 9)
WORD PROBLEM

45. Find the area of the
green shaded region.
x + 3
x-3 x
x+2