Polynomials and Factoring

Polynomials and
Factoring
The basic building blocks of
algebraic expressions


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


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


What about x = -5?
Be careful with the negative! Use ( )!
x2 – 4x + 1
= (-5)2 – 4(-5) + 1
= 46

You try a couple
Use the same expression but
let
x = 2 and
x = -1


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


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.


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 520x2y2
(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


trinomial – is the sum of three monomials.
It has three unlike terms. (tri implies three).

x2 + 2x + 1, 3x2 – 4x + 10, 2x + 3y
+2
The ending of these
words “nomial” is Greek
for “part”.

polynomial – is a monomial or the sum (+)
or difference (-) of one or more terms.
(poly implies many).
x2 + 2x,
8

3x3 + x2 + 5x + 6,

4x + 6y +

• 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)

Constants like 12 are monomials
since they can be written as 12x0
= 12 · 1 = 12 where the variable
is x0.


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
b) 7x2y3

degree: 1 ¾x = ¾x . The exponent is 1.
degree: 5 The exponents are 2 and 3. Their sum is
1

5.

c) -4

degree: 0

The degree of a nonzero constant is 0.


Here’s a polynomial

2x3 – 5x2 + x + 9
Each one of the little product things is a “term”.

2x3 – 5x2 + x + 9
term

term

term

term

So, this guy has 4 terms.

2x3 – 5x2 + x + 9
The coefficients are the numbers in front of the letters.

2x3 - 5x2 + x + 9
NEXT

2

5
Remember
x=1·x

1

9
We just pretend
this last guy has a
letter behind him.


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:
NEXT
7x3 + 5x2 – 2x + 4


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 guy has 5
letters…
The degree is 5.

This is a 7th degree polynomial:
6mn2 + m3n4 + 8
This guy has 7
letters…
The degree is 7

NEXT

This is a 1st degree polynomial
3x – 2
This guy has 1
letter…
The degree is
1.
By the way, the
coefficients
don’t have
anything to do
with the degree.

What about this dude?
8

This guy has no
letters…
The degree is 0.

How many letters does he have? ZERO!
So, he’s a zero degree polynomial
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!


3x4 + 5x2 – 7x + 1

term

term

term

term

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.

Once you simplify a polynomial by
combining like terms, you can name
the polynomial based on degree or
number of monomials it contains.


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
3x4 + 5x4 + 2x2 – 4
8x4 + 2x2 – 4

4th degree trinomial

Place terms in order.
Combine like terms.


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


Adding and
Subtracting
Polynomials
The sum or difference


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.


Addition of
Polynomials
Method 1 (vertically)

You can rewrite each
polynomial, inserting a zero
placeholder for the “missing”
term.

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)
Example 2
Use a zero placeholder

Simplify each sum

• (12m2 + 4) + (8m2 + 5)
• (t2 – 6) + (3t2 + 11)

Remember
Use a zero as a placeholder
for the “missing” term.

• (9w3 + 8w2) + (7w3 + 4)
• (2p3 + 6p2 + 10p) + (9p3 + 11p2 + 3p )

Word Problem


Find the perimeter of
each figure

9c - 10

-6

8x - 2

17x

5c +
2

5x
+
1

Recall that the
perimeter of a figure is
the sum of all the
sides.

9x


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


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



Simplify each
subtraction
• (17n4 + 2n3) – (10n4 + n3)
• (24x5 + 12x) – (9x5 + 11x)
• 6c – 5
-(4c + 9)
10)

2b + 6
-(b + 5)

7h 2 + 4h - 8
-(3h2 – 2h +