Download Degree of a Monomial

Factoring Part 1
Monomial:
Monomial:
A Number, a Variable or the
product of numbers and
variables with whole number
exponents.
Monomial:
A Number, a Variable or the
product of numbers and
variables with whole number
exponents.
Examples:
Monomial:
A Number, a Variable or the
product of numbers and
variables with whole number
exponents.
Examples: 10
3p
2
x
-5y
2xyz
Monomial: A Number, a Variable or
the product of numbers and
variables w/ whole number
exponents.
Are these monomials? Why or why
not?
1 2
ab
2
5x
4
0
n
x5
x
1
Monomial: A Number, a Variable or
the product of numbers and
variables w/ whole number
exponents.
Are these monomials? Why or why
not?
x5
1 2
ab yes
2
5x
4
0
n
x
1
Monomial: A Number, a Variable or
the product of numbers and
variables w/ whole number
exponents.
Are these monomials? Why or why
not?
x5
1 2
ab yes
2
5x
4
0 yes
n
x
1
Monomial: A Number, a Variable or
the product of numbers and
variables w/ whole number
exponents.
Are these monomials? Why or why
not?
x5
1 2
ab yes
2
n
4 no
0 yes
1
5x
x
Monomial: A Number, a Variable or
the product of numbers and
variables w/ whole number
exponents.
Are these monomials? Why or why
not?
x5
1 2
ab yes
2
n
not have a variable
4 no canexponent
0
1
5x yes
x
Monomial: A Number, a Variable or
the product of numbers and
variables w/ whole number
exponents.
Are these monomials? Why or why
not?
x5
1 2
ab yes
no
2
n
not have a variable
4 no canexponent
0
1
5x yes
x
Monomial: A Number, a Variable or the
product of numbers and variables w/
whole number exponents.
Are these monomials? Why or why not?
yes
no this is
1 2
ab
sum not a
2product
n
4 no can not have a variable
x5
5x
0
exponent
yes
x
1
Monomial: A Number, a Variable or the
product of numbers and variables w/
whole number exponents.
Are these monomials? Why or why not?
no this is
x5
1 2
ab yes
2
0
5x yes
4
n no
sum not a product
can not have a variable
exponent
x
1
no
Monomial: A Number, a Variable or the
product of numbers and variables w/
whole number exponents.
Are these monomials? Why or why not?
no this is
1 2 yes
ab
2
5x
0 yes
x5
4
n
sum not a product
no can not have a variable
exponent
x
1 no
can not have a negative exponent
Degree of a Monomial:
Degree of a Monomial:
The sum of the variable
exponents of the monomial.
Degree of a Monomial:
The sum of the variable
exponents of the monomial.
Example: 5x2y has degree ___
Degree of a Monomial:
The sum of the variable
exponents of the monomial.
2
Example: 5x y has degree 3
Degree of a Monomial:
Find the degree of the
monomials given.
2
3
2
3 ab c
3
-4x y
Degree of a Monomial:
Find the degree of the
monomials given.
2
3
2
3 ab c
3
-4x y
degree 6
Degree of a Monomial:
Find the degree of the
monomials given.
2
3
2
3 ab c
degree 6
Why not degree 8?????
3
-4x y
Degree of a Monomial:
Find the degree of the
monomials given.
2
3
2
3 ab c
degree 6
Why not degree 8????? You do NOT count the exponent of
the number 3, you only count the exponents of the variables.
3
-4x y
Degree of a Monomial:
Find the degree of the
monomials given.
2
3
2
3 ab c
3
-4x y
degree 6
degree 4
Polynomials:
Polynomials:
Are monomials or the sum or
difference of monomials.
Polynomials:
Are monomials or the sum or
difference of monomials.
Are these polynomials?
x+5
2x2 + 9x
3x-2y – 4x
Polynomials:
Are monomials or the sum or
difference of monomials.
Are these polynomials?
x + 5 yes, because each term
IS a monomial
2x2 + 9x
-2
3x y – 4x
Polynomials:
Are monomials or the sum or
difference of monomials.
Are these polynomials?
x + 5 yes, because each term
IS a monomial
2x2 + 9x yes, because each term
IS a monomial
3x-2y – 4x
Polynomials:
Are monomials or the sum or
difference of monomials.
Are these polynomials?
x + 5 yes, because each term
IS a monomial
2x2 + 9x yes, because each term
IS a monomial
3x-2y – 4x no, because you can not
have a negative exponent.
Polynomials:
Are monomials or the sum or difference
of monomials.
Are these polynomials?
x+5
yes, because each term
IS a monomial
2x2 + 9x yes, because each term
IS a monomial
3x-2y – 4x no, because you can not
have a negative exponent. If it is
NOT a monomial, it can not be a
polynomial!
Other types of Polynomials:
Binomials are polynomials that have
____ terms.
Other types of Polynomials:
Binomials are polynomials that have
TWO terms.
Other types of Polynomials:
Binomials are polynomials that have
TWO terms.
TERMS are separated by ________
and/or _________ signs.
Other types of Polynomials:
Binomials are polynomials that have
TWO terms.
TERMS are separated by addition
and/or subtraction signs.
Other types of Polynomials:
Binomials are polynomials that have
TWO terms.
TERMS are separated by addition
and/or subtraction signs.
Examples of binomials:
x2 - 5
4yz3 + 3y2z2
Other types of Polynomials:
Binomials are polynomials that have
TWO terms.
TERMS are separated by ________
and/or _________ signs.
Examples of binomials:
x2 - 5
4yz3 + 3y2z2
NOT a binomial:
3y-2 + 5
Other types of Polynomials:
Binomials are polynomials that have
TWO terms.
TERMS are separated by ________
and/or _________ signs.
Examples of binomials:
x2 - 5
4yz3 + 3y2z2
NOT a binomial:
3y-2 + 5
WHY?
Other types of Polynomials:
Binomials are polynomials that have
TWO terms.
TERMS are separated by ________
and/or _________ signs.
Examples of binomials:
x2 - 5
4yz3 + 3y2z2
NOT a binomial:
3y-2 + 5
WHY? You can not have negative exponents.
Other types of Polynomials:
Trinomials are polynomials that have
THREE terms.
Other types of Polynomials:
Trinomials are polynomials that have
THREE terms.
Example:
-3x3 + 5x2 - 10
Degree of Polynomials:
Degree of Polynomials:
The greatest degree term.
Degree of Polynomials:
The greatest degree term.
What does that mean?
Degree of Polynomials:
The greatest degree term.
What does that mean?
Find the degree of each term of
the polynomial. Pick the
biggest number.
Degree of Polynomials:
The greatest degree term.
What does that mean?
Find the degree of each term of
the polynomial. Pick the
biggest number.
Example: 2x3y – xy4 + 9x3y4
Degree of Polynomials:
The greatest degree term.
What does that mean?
Find the degree of each term of
the polynomial. Pick the
biggest number.
Example: 2x3y – xy4 + 9x3y4
Degree:
Degree of Polynomials:
The greatest degree term.
What does that mean?
Find the degree of each term of
the polynomial. Pick the
biggest number.
Example: 2x3y – xy4 + 9x3y4
Degree: 4
Degree of Polynomials:
The greatest degree term.
What does that mean?
Find the degree of each term of
the polynomial. Pick the
biggest number.
Example: 2x3y – xy4 + 9x3y4
Degree: 4
5
Degree of Polynomials:
The greatest degree term.
What does that mean?
Find the degree of each term of
the polynomial. Pick the
biggest number.
Example: 2x3y – xy4 + 9x3y4
Degree: 4
5
7
Example: 2x3y – xy4 + 9x3y4
Degree: 4
5
7
The degree of the
polynomial is 7.
Rewrite the polynomial in
descending order:
Rewrite the polynomial in
descending order:
Example 1:
7+
4
2x
– 4x
Rewrite the polynomial in
descending order:
Example 1:
7+
– 4x
The highest degree term
must go first.
4
2x
Rewrite the polynomial in
descending order:
Example 1:
7+
– 4x
The highest degree term
must go first.
4
2x
4
2x
Rewrite the polynomial in
descending order:
Example 1:
7+
– 4x
The highest degree term
must go first.
4
2x – 4x
4
2x
Rewrite the polynomial in
descending order:
Example 1:
7+
– 4x
The highest degree term
must go first.
4
2x – 4x + 7
4
2x
Rewrite the polynomial in
descending order:
Example 1:
7+
4
2x
– 4x
The highest degree term must
4
go first. 2x – 4x + 7
Be careful to keep the negative
signs with the correct term.
Leading Coefficient
of a polynomial:
Leading Coefficient
of a polynomial:
Is the coefficient of the highest
degree term.
Leading Coefficient
of a polynomial:
Is the coefficient of the highest
degree term.
What does coefficient mean?????
Leading Coefficient
of a polynomial:
Is the coefficient of the highest degree term.
What does coefficient mean?????
The number in front of the variable, attached
by multiplication.
Leading Coefficient
of a polynomial:
The number in front of the variable, attached
by multiplication.
CHECKPOINT – page 189
1. 5x + 13 + 8x3
Leading Coefficient
of a polynomial:
The number in front of the variable, attached
by multiplication.
CHECKPOINT
1. 5x + 13 + 8x3
= 8x3 + 5x + 13
CHECKPOINT
2. 4y4 – 7y5 + 2y
CHECKPOINT
1. 5x + 13 + 8x3
=
8x3 + 5x + 13
2. 4y4 – 7y5 + 2y
=
-7y5 + 4y4 + 2y
Example 2
a.
b.
c.
d.
e.
Example 2
a. Yes
b.
c.
d.
e.
Example 2
a. Yes
b.
c.
d.
e.
o degree, monomial
Example 2
a. Yes
b. No
c.
d.
e.
o degree, monomial
Example 2
a. Yes
b. No
c.
d.
e.
o degree, monomial
you can’t have negative
exponents.
Example 2
a. Yes
o degree, monomial
b. No
you can’t have negative
exponents.
c. Yes
d.
e.
Example 2
a. Yes
o degree, monomial
b. No
you can’t have negative
exponents.
c. Yes
degree 3, binomial
d.
e.
Example 2
a. Yes
o degree, monomial
b. No
you can’t have negative
exponents.
c. Yes
degree 3, binomial
d. Yes
e.
Example 2
a. Yes
o degree, monomial
b. No
you can’t have negative
exponents.
c. Yes
degree 3, binomial
d. Yes
degree 4, trinomial
e.
Example 2
a. Yes
o degree, monomial
b. No
you can’t have negative
exponents.
c. Yes
degree 3, binomial
d. Yes
degree 4, trinomial
e. No
Example 2
a. Yes
b. No
c. Yes
d. Yes
e. No
o degree, monomial
you can’t have negative
exponents.
degree 3, binomial
degree 4, trinomial
you can’t have a variable
exponent
CHECKPOINT
3. 4x – x7 + 5x3
CHECKPOINT
3. 4x – x7 + 5x3
yes
CHECKPOINT:
3. 4x – x7 + 5x3
yes
degree 7
CHECKPOINT:
3. 4x – x7 + 5x3
yes
degree 7
trinomial
CHECKPOINT:
4. v3 + v-2 + 2v
CHECKPOINT:
4. v3 + v-2 + 2v
No
because you can have a negative exponent
Example 3
a. (4x3 + x2 – 5) + (7x + x3 – 3x2)
Example 3
a. (4x3 + x2 – 5) + (7x + x3 – 3x2)
Horizontal formal………
Example 3
(4x3 + x2 – 5) + (7x + x3 – 3x2)
Horizontal formal………Add like terms
Example 3
a. (4x3 + x2 – 5) + (7x + x3 – 3x2)
Horizontal formal………Add like terms and
write answer in descending order of
exponents.
5x3
Example 3
a. (4x3 + x2 – 5) + (7x + x3 – 3x2)
Horizontal formal………Add like terms and
write answer in descending order of
exponents.
5x3 – 2x2
Example 3
a. (4x3 + x2 – 5) + (7x + x3 – 3x2)
Horizontal formal………Add like terms and
write answer in descending order of
exponents.
5x3 – 2x2 +7x
Example 3
a. (4x3 + x2 – 5) + (7x + x3 – 3x2)
Horizontal formal………Add like terms and
write answer in descending order of
exponents.
5x3 – 2x2 +7x - 5
Example 3
b. (x2 + x + 8) + (x2 – x – 1)
Example 3
b. (x2 + x + 8) + (x2 – x – 1)
= 2x2
Example 3
b. (x2 + x + 8) + (x2 – x – 1)
= 2x2 + 7
Example 4
a. (4z2 – 3) - (-2z2 + 5z – 1)
Horizontal formal………
Example 4
a. (4z2 – 3) - (-2z2 + 5z – 1)
Horizontal formal………Distribute the
subtract sign to EVERYTHING inside the
back set of parenthesis.
Example 4
a. (4z2 – 3) - (-2z2 + 5z – 1)
Horizontal formal………Distribute the subtract sign
to EVERYTHING inside the back set of
parenthesis.
(4z2 – 3) + (+2z2 - 5z + 1)
Example 4
a. (4z2 – 3) - (-2z2 + 5z – 1)
Horizontal formal………Distribute the subtract sign
to EVERYTHING inside the back set of
parenthesis.
= (4z2 – 3) + (+2z2 - 5z + 1)
= 6z2
Example 4
a. (4z2 – 3) - (-2z2 + 5z – 1)
Horizontal formal………Distribute the subtract sign
to EVERYTHING inside the back set of
parenthesis.
= (4z2 – 3) + (+2z2 - 5z + 1)
= 6z2 - 5z
Example 4
a. (4z2 – 3) - (-2z2 + 5z – 1)
Horizontal formal………Distribute the subtract sign
to EVERYTHING inside the back set of
parenthesis.
= (4z2 – 3) + (+2z2 - 5z + 1)
= 6z2 - 5z -2
Example 4
b. (3x2 + 6x – 4) – (x2 – x – 7)
Horizontal formal………You do this one.
Example 4
b. (3x2 + 6x – 4) – (x2 – x – 7)
Horizontal formal………
= 2x2 + 7x + 3
SKIP TO PAGE 192
2
3
2x (x
= 2x5
–
2
5x
+ 3x – 7)
–
= 2x5 – 10x4
2
3
2x (x
2
5x
+ 3x – 7)
–
+ 3x – 7)
= 2x5 – 10x4 + 6x3
2
3
2x (x
2
5x
–
+ 3x – 7)
= 2x5 – 10x4 + 6x3 – 14x2
2
3
2x (x
2
5x
SKIP TO EXAMPLE 3
EXAMPLE 3
(2c + 7)(c – 9) =
F
O
I
L
EXAMPLE 3
(2c + 7)(c – 9) =
F stands for First
O
I
L
EXAMPLE 3
(2c + 7)(c – 9) =
F stands for First
O
I
L
EXAMPLE 3
(2c + 7)(c – 9) =
F stands for First
O stands for Outer
I
L
EXAMPLE 3
(2c + 7)(c – 9) =
F stands for First
O stands for Outer
I
L
EXAMPLE 3
(2c + 7)(c – 9) =
F stands for First
O stands for Outer
I stands for Inner
L
EXAMPLE 3
(2c + 7)(c – 9) =
F stands for First
O stands for Outer
I stands for Inner
L
EXAMPLE 3
(2c + 7)(c – 9) =
F stands for First
O stands for Outer
I stands for Inner
L stands for Last
EXAMPLE 3
(2c + 7)(c – 9) =
F stands for First
O stands for Outer
I stands for Inner
L stands for Last
(2c + 7)(c – 9) =
F
O
I
L
(2c + 7)(c – 9) =
F 2c2
O
I
L
(2c + 7)(c – 9) =
F 2c2
O
I
L
(2c + 7)(c – 9) =
F 2c2
O
-18c
I
L
(2c + 7)(c – 9) =
F 2c2
O
-18c
I
L
(2c + 7)(c – 9) =
F 2c2
O
-18c
I
7c
L
(2c + 7)(c – 9) =
F 2c2
O
-18c
I
7c
L
(2c + 7)(c – 9) =
F 2c2
O
-18c
I
7c
L
-63
(2c + 7)(c – 9) =
F 2c2
O
-18c
I
7c
L
Put it all together.
-63
(2c + 7)(c – 9) =
F 2c2
O
-18c
I
7c
L
-63
Put it all together.
= 2c2 – 11c - 63
(m + 3)(5m – 4)
F
O
I
L