Download 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
polynomial – is a monomial or the sum (+) or
difference (-) of one or more terms.
(poly implies many).
x2 + 2x,
3x3 + x2 + 5x + 6,
The ending of these
words “nomial” is Greek
for “part”.
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)
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
c) -4
degree: 1
degree: 5
degree: 0
¾x = ¾x1. The exponent is 1.
The exponents are 2 and 3. Their sum is 5.
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
2
5
Remember
x=1·x
1
NEXT
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.
Polynomial
Degree
7x + 4
3x + 2x + 1
4x3
9x4 + 11x
5
1
2
3
4
0
2
Name using
Degree
Linear
Quadratic
Cubic
Fourth degree
Constant
Once you simplify a polynomial by
combining like terms, you can name the
polynomial based on degree or number of
monomials it contains.
Number of
Terms
2
3
1
2
1
Name using
number of
terms
Binomial
Trinomial
Monomial
Binomial
monomial
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
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
You can rewrite each polynomial,
inserting a zero placeholder for the
“missing” term.
Method 1 (vertically)
Line up like terms. Then add the coefficients.
4x2 + 6x + 7
2x2 – 9x + 1
6x2 – 3x + 8
Method 2 (horizontally)
-2x3 + 2x2 – 5x + 3
0 + 5x2 + 4x - 5
-2x3 + 7x2 – x - 2
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
5c + 2
Recall that the perimeter
of a figure is the sum of
all the sides.
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
-(x3 – 8x2 + 0 + 11)
Remember,
subtraction is adding
the opposite.
2x3 + 5x2 – 3x
-x3 + 8x2 + 0 - 11
x3 +13x2 – 3x - 11
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)
2b + 6
-(b + 5)
7h2 + 4h - 8
-(3h2 – 2h + 10)
Multiplying and Factoring
Using the Distributive Property
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.
4x
2x
A = lw
A = (4x)(2x)
****************************
x+x+x+x
x
+
x
The rectangle above shows the example that
NEXT
4x = x + x + x + x and 2x = x + x
We can further divide the rectangle into
squares with side lengths of x.
x +x+x+ x
x
+
x
Since each side of the squares
are x, then x · x = x2
x+x+x+x
x
+
x
x2
x2
x2
x2
x2
x2
x2
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
Commutative Property
Associative Property
NEXT
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;
(4x)(2x) = (4 · 2)(x · x ) =
• Calculate the product of the numerical
coefficients; and
(4 · 2) = 8
• Use the properties of exponents to simplify the
variable product.
(x · x) = x1 · x1 = x1+1 = x2
Therefore (4x)(2x) = 8x2
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)
Remember,
Multiply powers with the same base:
35 · 34 = 35 + 4 = 39
-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.
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.
Factoring a Monomial
from a Polynomial
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.
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 each polynomial.
a) 5v5 + 10v3
b) 3t2 – 18
c) 4b3 – 2b2 – 6b
d) 2x4 + 10x2 – 6x
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
To factor a polynomial
completely, you must factor
until there are no common
factors other than 1.
Step 2
Factor out the GCF
3x3 – 12x2 + 15x
= 3x(x2) + 3x(-4x) + 3x(5)
= 3x(x2 – 4x + 5)
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.
Multiplying Binomials
Using the infamous FOIL method
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.
Now Distribute 2x and 3
Distribute x + 4
Simplify:
(2x + 3)(x + 4)
(2x + 3)(x + 4) =
2x(x + 4) + 3(x + 4) =
2x2 + 8x + 3x + 12 =
2x2 + 11x + 12
Simplify each
product.
a) (6h – 7)(2h + 3)
b) (5m + 2)(8m – 1)
c) (9a – 8)(7a + 4)
d) (2y – 3)(y + 2)
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
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
Applying
Multiplication of
Polynomials.
area of outer rectangle =
(2x + 5)(3x + 1)
area of orange rectangle =
Find the area of the
shaded (beige) region.
Simplify.
x(x + 2)
area of shaded region
= area of outer rectangle – area of
orange portion
2x + 5
x+2
3x + 1
x
Use the FOIL method to
simplify (2x + 5)(3x + 1)
(2x + 5)(3x + 1) – x(x + 2) =
6x2 + 15x + 2x + 5 – x2 – 2x =
6x2 – x2 + 15x + 2x – 2x + 5 =
5x2 + 17x + 5
Use the Distributive Property
to simplify –x(x + 2)
Find the area of the
shaded region.
Simplify.
Find the area of the green shaded region. Simplify.
6x + 2
5x + 8
5x
x+6
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.)
(4x2
Remember multiplying
whole numbers.
312
x 23
936
624
7176
Simplify
+ 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
Multiply using the
horizontal method.
Method 2 (horizontal)
Drawing arrows
between terms can
help you identify all six
products.
(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
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
Find the area of the
green shaded region.
x+3
x+2
x
x-3