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Polynomials
No description
by
TweetRob Frederick
on 23 January 2012Transcript of Polynomials
( a b)(a c )(6b )
(
a
b
)(
a
c
)(
6
b
)
( 6)
(a a )
(b b )
(c )
1
2
_
Polynomials
Bell Ringer
1.) You want to build a playhouse in your backyard. Your blueprints show that the playhouse is 10 feet x 13 feet. You want to change the dimensions as shown below.
For which value of x can you have a A.) x = 2 ft playhouse with an area of 132 square feet?
B.) x = 3 ft
C.) x = 4 ft
D.) x = 5 ft
2.) Miguel earns a 14% commission on his sales in addition to a salary of $500 a week. His earnings can be modeled by the equation
p = 0.15s + 500
. What restrictions on the values of
p
and
s
best fit this situation?
F.)
p > 0
,
s
can be any value
G.)
s > 0
,
p
can be any value
H.)
s > 500
,
p > 0
J.)
s > 0
,
p > 500
10 ft
13 ft
x
13  x






Objectives
A.) Classify Polynomials
B.) Rewrite Polynomials in Standard Form
C.) Evaluate Polynomials Expressions
D.) Add and Subtract Polynomials
E.) Multiply Polynomials (including finding Special Products)
PASS 1.2a: Simplify and evaluate linear, absolute value, rational and radical expressions.
PASS 1.2b: Simplify polynomials by adding, subtracting or multiplying.
Let's talk about what some of these words mean...
First...
A
monomial
is any number, variable, or a product of numbers and variables with whole number exponents.
The
degree of a monomial
is the sum of the exponents of the variables. A constant has a degree of 0.
Monomials
NOT Monomials
5, x, 7xy, 0.5x
4
0.3x , 4x  y,
2
x
2
2
_
Classifying Polynomials
A few more words...
On the other hand, a
polynomial
is a monomial or a sum or difference of monomials.
While
degree of a polynomial
is the degree of the term with the greatest degree.
So what would be the degree of these polynomials?
4x  18x
5
0.5x y + 0.25xy + 0.75
2
6x + 9x  x + 3
4
2
The degree of the polynomial is the degree of the largest term.
4x 18x
degree: 1 degree: 5
So in this case, the degree of the polynomial is:
5
5
0.5x y
degree: 3
0.25xy
degree: 2
0.75
degree 0
The degree of this polynomial:
3
2
degree:
*
"write it down" symbol of the day
*
*
*
*
And then...
some charts...
Polynomials named by degree
Polynomials named by
number of terms
Some polynomials have special names
because of degree or number of terms...
Degree
0
1
2
3
4
5
6 or more
Name
constant
linear
quadratic
cubic
quartic
quintic
6th degree, 7th degree, etc.
Terms
1
2
3
4 0r more
Name
monomial
binomial
trinomial
polynomial
*
*
The greatly talked about "standard form"...
Finally...
The
standard form of a polynomial
(that contains only one variable) is written with the terms in order from greatest degree from least degree.
*
When written in standard form, the coefficient of the first term is called the
leading coefficient
.
So, 20x  4x + 2  x becomes 4x  x + 20x + 2 in standard form.
degree: 1 3 0 2 3 2 1 0
3
3
2
2
The leading coefficient here: 4.
*
Adding/Subtracting Polynomials
To add or subtract monomials,
we can follow three steps.
Adding/Subtracting Monomials
15m + 6m + 2m
15m
+ 6m
+ 2m
1.) Identify like terms
.
15m + 2m
+ 6m
2.) Rearrange (like terms together)
.
17m
+ 6m
3.) Combine like terms
.
3
3
2
2
2
2
3
3
3
3
3
DON'T FORGET:
terms with
different variables
OR
different exponents
ARE
NOT LIKE TERMS
....and cannot ever be combined.
Add or Subtract:
Practice w/ Monomials...
Answers:
1.) 3x + 5  7x + 12
2.) 2x y  x y  x y
2
2
2
2
2
1.) 4x + 17
2
2.) 0
Just like adding numbers, polynomials can be added in vertical or horizontal form:
Adding/Subtracting Polynomials
V
E
R
T
I
C
A
L
H O R I Z O N T A L
In vertical form,
align like terms and add:
In horizontal form,
regroup and then combine like terms:
5x
+ 4x
+ 1
+
2x
+ 5x
+ 2
7x
+ 9x
+ 3
(
5x
+ 4x
+ 1
) + (
2x
+ 5x
+ 2
)
(5x + 2x )
+
(4x + 5x)
+ (
1 + 2
)
7x
+ 9x
+ 3
2
2
2
2
2
2
2
Add or subtract:
Practice w/ Polynomials
1.) (2x  x) + (x + 3x 1)
2.) (2ab + b) + (2ab + a)
3.) (a  2a)  (3a  3a + 1)
Answers:
1.) 3x + 2x  1
2.) b + a
3.) 2a + a  1
2
2
4
4
4
2
*When subtracting polynomials,
remember to first distribute the negative to all of the terms
appropriate polynomial...then it's just like adding two polynomials...
*
*
*
Multiplying Polynomials
2
Bell Ringer
So, now we'll talk about how to multiply polynomials.
We've added and subtracted...
All the rules we've learned so far still apply:
1.) multiplication of constants (numbers)
2.) multiplication rules for exponents
3.) the distributive property
REMEMBER
Quick reminder on
positives and negatives:
1.) When two signs AGREE the result is ALWAYS POSITIVE.
3
(
4
) =
12
3
(
4
) =
12
2.) When two signs DISAGREE the result is ALWAYS NEGATIVE.
3
(
4
) =
12
3
(
4
) =
12
When multiplying, the following two rules apply to exponents
Multiplication Rules
for Exponents
1.) When multiplying two powers with the SAME BASE,
exponents are ADDed.
a (a ) = a 3 (3 ) = 3
*If the bases are not the same, this does NOT apply.
4
7
11
8
3
11
2.) When there is a POWER OF A POWER,
MULTIPLY the exponents.
(a ) = a (3 ) = 3
*Notice ONLY the exponents change,
the base does not.
4
7
8
3
28
24
When multiplying, make sure to distribute,
or "pass out," to everything in the parenthesis.
(Just like passing out papers in class and making
sure everyone gets a copy...)
The Distributive Property
a(b + c) = ab + ac
2(x + 3) = 2x + 6
*This applies no matter how many
terms are in the parenthesis...
Multiplying Monomials
Let's put those
rules together now...
(5x )(4x )
(
5
x
)(
4
x
)
(5 4)
(x x )
20
x
*There's really only one "rule"
other than the multiplication rules:
Group factors with like bases together,
and then multiply.
(3x y )(4xy )
(
3
x
y
)(
4
x
y
)
(3 4)
(x x)
(y y )
12
x
y
Now you try one:
( a b)(a c )(6b )
2
3
2
2
5
3
3
.
.
.
.
7
2
5
3
3
3
4
2
5
2
5
.
1
2
_
3
2
2
2
You should have worked it out
and gotten he following solution:
1
2
_
3
2
2
2
3
2
2
2
2
1
_
3
2
2
2
.
.
.
3
a
b
c
5
3
2
Multiplying a Polynomial by a Monomial
A Second Example...
*For this, we need to use the distributive property...
5(2x + x + 4)
5
(2x + x + 4)
(5)
2x
+
(5)
x
+
(5)
4
10x
+
5x
+
20
DISTRIBUTE!
2
2
2
2
PRACTICE!
*Don't forget to group like bases.
Multiply...
4a(a b + 2b )
2
2
You should have gotten:
Solution...
*Check your multiplication rules
if you made a mistake...
4a b + 8ab
3
2
Multiplying Binomials by Binomials
The FOIL Method...
To multiply binomials, we use the distributive property
more than once...this is called the FOIL method.
irst
uter
nner
ast
(x + 3)(x + 2)
1.) Multiply the
First
terms. (
x
+ 3)(
x
+ 2)
x
2.) Multiply the
Outer
terms. (
x
+ 3)(x +
2
)
2x
3.) Multiply the
Inner
terms. (x +
3
)(
x
+ 2)
3x
4.) Multiply the
Last
terms. (x +
3
)(x +
2
)
6
Then we simplify by adding like terms to get:
x + 5x + 6
2
2
is a way to visualize FOIL and remember it more easily.
The FOIL Face...
(x + 3)(x + 2)
Multiply. Don't forget to take care of each part of FOIL...
PRACTICE FOIL
(x + 2)(x  5)
(x + 5)
2
You should have gotten:
PRACTICE FOIL SOLUTIONS
(x + 2)(x  5) = x  3x  10
(x + 5) = x + 10x + 25
2
2
2
If you got something else:
1.) make sure you didn't leave out
part of FOIL...
2.) check your multiplication rules...
And finally...
The Box Method
The Vertical Method
For everything else, you can always use one of these two
methods for multiplying polynomials by polynomials.
Multiplying Polynomials
(2x + 10x  6)(5x + 3)
(2x + 10x  6)(5x + 3)
2
2
2x 10x 6
5x 10x 50x 30x
3 6x 30x 18
2x 10x 6
5x
3
1.) Rewrite the polynomials on the width/length of a box. Make sure every term is represented in the correct polynomial.
2.) Write the product of the monomials in each row and column.
2
2
3
2
2
10x + 6x + 50x + 30x  30x  18
3.) Add like terms.
SOLUTION: 10x + 56x  18
3
2
2
3
2
SOLUTION: 10x + 56x  18
3
2
2x + 10x  6
5x
+ 3
6x + 30x  18
10x + 50x  30x
10x + 56x  18
x
1.) Rewrite the polynomials and line them up just like you would whole numbers when multiplying.
2.) Multiply all of terms in the top polynomial by the smallest degree term in the bottom polynomial.
3.) Multiply all of the terms in the top polynomial by the next degree term in the bottom polynomial. (repeat if nec.)
4.) Add like terms vertically.
Simplify if necessary.
3
2
2
2
2
3
*
*
*
*
*
*
Use any method you prefer.
Multiply...
(3x  4)(2x + 5x 6)
3
You should have gotten:
6x + 8x + 15x 38x + 24
3
2
4
Full transcript(
a
b
)(
a
c
)(
6
b
)
( 6)
(a a )
(b b )
(c )
1
2
_
Polynomials
Bell Ringer
1.) You want to build a playhouse in your backyard. Your blueprints show that the playhouse is 10 feet x 13 feet. You want to change the dimensions as shown below.
For which value of x can you have a A.) x = 2 ft playhouse with an area of 132 square feet?
B.) x = 3 ft
C.) x = 4 ft
D.) x = 5 ft
2.) Miguel earns a 14% commission on his sales in addition to a salary of $500 a week. His earnings can be modeled by the equation
p = 0.15s + 500
. What restrictions on the values of
p
and
s
best fit this situation?
F.)
p > 0
,
s
can be any value
G.)
s > 0
,
p
can be any value
H.)
s > 500
,
p > 0
J.)
s > 0
,
p > 500
10 ft
13 ft
x
13  x






Objectives
A.) Classify Polynomials
B.) Rewrite Polynomials in Standard Form
C.) Evaluate Polynomials Expressions
D.) Add and Subtract Polynomials
E.) Multiply Polynomials (including finding Special Products)
PASS 1.2a: Simplify and evaluate linear, absolute value, rational and radical expressions.
PASS 1.2b: Simplify polynomials by adding, subtracting or multiplying.
Let's talk about what some of these words mean...
First...
A
monomial
is any number, variable, or a product of numbers and variables with whole number exponents.
The
degree of a monomial
is the sum of the exponents of the variables. A constant has a degree of 0.
Monomials
NOT Monomials
5, x, 7xy, 0.5x
4
0.3x , 4x  y,
2
x
2
2
_
Classifying Polynomials
A few more words...
On the other hand, a
polynomial
is a monomial or a sum or difference of monomials.
While
degree of a polynomial
is the degree of the term with the greatest degree.
So what would be the degree of these polynomials?
4x  18x
5
0.5x y + 0.25xy + 0.75
2
6x + 9x  x + 3
4
2
The degree of the polynomial is the degree of the largest term.
4x 18x
degree: 1 degree: 5
So in this case, the degree of the polynomial is:
5
5
0.5x y
degree: 3
0.25xy
degree: 2
0.75
degree 0
The degree of this polynomial:
3
2
degree:
*
"write it down" symbol of the day
*
*
*
*
And then...
some charts...
Polynomials named by degree
Polynomials named by
number of terms
Some polynomials have special names
because of degree or number of terms...
Degree
0
1
2
3
4
5
6 or more
Name
constant
linear
quadratic
cubic
quartic
quintic
6th degree, 7th degree, etc.
Terms
1
2
3
4 0r more
Name
monomial
binomial
trinomial
polynomial
*
*
The greatly talked about "standard form"...
Finally...
The
standard form of a polynomial
(that contains only one variable) is written with the terms in order from greatest degree from least degree.
*
When written in standard form, the coefficient of the first term is called the
leading coefficient
.
So, 20x  4x + 2  x becomes 4x  x + 20x + 2 in standard form.
degree: 1 3 0 2 3 2 1 0
3
3
2
2
The leading coefficient here: 4.
*
Adding/Subtracting Polynomials
To add or subtract monomials,
we can follow three steps.
Adding/Subtracting Monomials
15m + 6m + 2m
15m
+ 6m
+ 2m
1.) Identify like terms
.
15m + 2m
+ 6m
2.) Rearrange (like terms together)
.
17m
+ 6m
3.) Combine like terms
.
3
3
2
2
2
2
3
3
3
3
3
DON'T FORGET:
terms with
different variables
OR
different exponents
ARE
NOT LIKE TERMS
....and cannot ever be combined.
Add or Subtract:
Practice w/ Monomials...
Answers:
1.) 3x + 5  7x + 12
2.) 2x y  x y  x y
2
2
2
2
2
1.) 4x + 17
2
2.) 0
Just like adding numbers, polynomials can be added in vertical or horizontal form:
Adding/Subtracting Polynomials
V
E
R
T
I
C
A
L
H O R I Z O N T A L
In vertical form,
align like terms and add:
In horizontal form,
regroup and then combine like terms:
5x
+ 4x
+ 1
+
2x
+ 5x
+ 2
7x
+ 9x
+ 3
(
5x
+ 4x
+ 1
) + (
2x
+ 5x
+ 2
)
(5x + 2x )
+
(4x + 5x)
+ (
1 + 2
)
7x
+ 9x
+ 3
2
2
2
2
2
2
2
Add or subtract:
Practice w/ Polynomials
1.) (2x  x) + (x + 3x 1)
2.) (2ab + b) + (2ab + a)
3.) (a  2a)  (3a  3a + 1)
Answers:
1.) 3x + 2x  1
2.) b + a
3.) 2a + a  1
2
2
4
4
4
2
*When subtracting polynomials,
remember to first distribute the negative to all of the terms
appropriate polynomial...then it's just like adding two polynomials...
*
*
*
Multiplying Polynomials
2
Bell Ringer
So, now we'll talk about how to multiply polynomials.
We've added and subtracted...
All the rules we've learned so far still apply:
1.) multiplication of constants (numbers)
2.) multiplication rules for exponents
3.) the distributive property
REMEMBER
Quick reminder on
positives and negatives:
1.) When two signs AGREE the result is ALWAYS POSITIVE.
3
(
4
) =
12
3
(
4
) =
12
2.) When two signs DISAGREE the result is ALWAYS NEGATIVE.
3
(
4
) =
12
3
(
4
) =
12
When multiplying, the following two rules apply to exponents
Multiplication Rules
for Exponents
1.) When multiplying two powers with the SAME BASE,
exponents are ADDed.
a (a ) = a 3 (3 ) = 3
*If the bases are not the same, this does NOT apply.
4
7
11
8
3
11
2.) When there is a POWER OF A POWER,
MULTIPLY the exponents.
(a ) = a (3 ) = 3
*Notice ONLY the exponents change,
the base does not.
4
7
8
3
28
24
When multiplying, make sure to distribute,
or "pass out," to everything in the parenthesis.
(Just like passing out papers in class and making
sure everyone gets a copy...)
The Distributive Property
a(b + c) = ab + ac
2(x + 3) = 2x + 6
*This applies no matter how many
terms are in the parenthesis...
Multiplying Monomials
Let's put those
rules together now...
(5x )(4x )
(
5
x
)(
4
x
)
(5 4)
(x x )
20
x
*There's really only one "rule"
other than the multiplication rules:
Group factors with like bases together,
and then multiply.
(3x y )(4xy )
(
3
x
y
)(
4
x
y
)
(3 4)
(x x)
(y y )
12
x
y
Now you try one:
( a b)(a c )(6b )
2
3
2
2
5
3
3
.
.
.
.
7
2
5
3
3
3
4
2
5
2
5
.
1
2
_
3
2
2
2
You should have worked it out
and gotten he following solution:
1
2
_
3
2
2
2
3
2
2
2
2
1
_
3
2
2
2
.
.
.
3
a
b
c
5
3
2
Multiplying a Polynomial by a Monomial
A Second Example...
*For this, we need to use the distributive property...
5(2x + x + 4)
5
(2x + x + 4)
(5)
2x
+
(5)
x
+
(5)
4
10x
+
5x
+
20
DISTRIBUTE!
2
2
2
2
PRACTICE!
*Don't forget to group like bases.
Multiply...
4a(a b + 2b )
2
2
You should have gotten:
Solution...
*Check your multiplication rules
if you made a mistake...
4a b + 8ab
3
2
Multiplying Binomials by Binomials
The FOIL Method...
To multiply binomials, we use the distributive property
more than once...this is called the FOIL method.
irst
uter
nner
ast
(x + 3)(x + 2)
1.) Multiply the
First
terms. (
x
+ 3)(
x
+ 2)
x
2.) Multiply the
Outer
terms. (
x
+ 3)(x +
2
)
2x
3.) Multiply the
Inner
terms. (x +
3
)(
x
+ 2)
3x
4.) Multiply the
Last
terms. (x +
3
)(x +
2
)
6
Then we simplify by adding like terms to get:
x + 5x + 6
2
2
is a way to visualize FOIL and remember it more easily.
The FOIL Face...
(x + 3)(x + 2)
Multiply. Don't forget to take care of each part of FOIL...
PRACTICE FOIL
(x + 2)(x  5)
(x + 5)
2
You should have gotten:
PRACTICE FOIL SOLUTIONS
(x + 2)(x  5) = x  3x  10
(x + 5) = x + 10x + 25
2
2
2
If you got something else:
1.) make sure you didn't leave out
part of FOIL...
2.) check your multiplication rules...
And finally...
The Box Method
The Vertical Method
For everything else, you can always use one of these two
methods for multiplying polynomials by polynomials.
Multiplying Polynomials
(2x + 10x  6)(5x + 3)
(2x + 10x  6)(5x + 3)
2
2
2x 10x 6
5x 10x 50x 30x
3 6x 30x 18
2x 10x 6
5x
3
1.) Rewrite the polynomials on the width/length of a box. Make sure every term is represented in the correct polynomial.
2.) Write the product of the monomials in each row and column.
2
2
3
2
2
10x + 6x + 50x + 30x  30x  18
3.) Add like terms.
SOLUTION: 10x + 56x  18
3
2
2
3
2
SOLUTION: 10x + 56x  18
3
2
2x + 10x  6
5x
+ 3
6x + 30x  18
10x + 50x  30x
10x + 56x  18
x
1.) Rewrite the polynomials and line them up just like you would whole numbers when multiplying.
2.) Multiply all of terms in the top polynomial by the smallest degree term in the bottom polynomial.
3.) Multiply all of the terms in the top polynomial by the next degree term in the bottom polynomial. (repeat if nec.)
4.) Add like terms vertically.
Simplify if necessary.
3
2
2
2
2
3
*
*
*
*
*
*
Use any method you prefer.
Multiply...
(3x  4)(2x + 5x 6)
3
You should have gotten:
6x + 8x + 15x 38x + 24
3
2
4