Transcript Adding and Subtracting Polynomials

Adding and Subtracting
Polynomials
Section 8-1
Goals
Goal
• To classify, add, and
subtract polynomials.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
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Monomial
Degree of a Monomial
Polynomial
Standard Form of a Polynomial
Degree of a Polynomial
Binomial
Trinomial
Definitions
A monomial is a number, a 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 degree 0.
Example: Degree of a
Monomial
Find the degree of each monomial.
A. 4p4q3
The degree is 7.
Add the exponents of the variables:
4 + 3 = 7.
B. 7ed
The degree is 2.
C. 3
The degree is 0.
Add the exponents of the variables:
1+ 1 = 2.
Add the exponents of the variables:
0 = 0.
Your Turn:
Find the degree of each monomial.
a. 1.5k2m
The degree is 3.
Add the exponents of the variables:
2 + 1 = 3.
b. 4x
The degree is 1.
Add the exponents of the variables:
1 = 1.
b. 2c3
The degree is 3.
Add the exponents of the variables:
3 = 3.
Like Terms
You can add or subtract monomials by
adding or subtracting like terms.
Like terms
The variables have the same powers.
4a3b2 + 3a2b3 – 2a3b2
Not like terms
The variables have different powers.
Add or subtract like terms by adding or subtracting
the coefficients of the like terms.
Example: Identify Like Terms
Identify the like terms in each polynomial.
A. 5x3 + y2 + 2 – 6y2 + 4x3
5x3 + y 2 + 2 – 6y2 + 4x 3
Identify like terms.
Like terms: 5x3 and 4x3, y2 and –6y2
B. 3a3b2 + 3a2b3 + 2a3b2 – a3b2
2 3
3 2
3a b + 3a b + 2a b – a b
3
2
3
2
Identify like terms.
Like terms: 3a3b2, 2a3b2, and –a3b2
Example: Identify Like Terms
Identify the like terms in the polynomial.
C. 7p3q2 + 7p2q3 + 7pq2
7p3q2 + 7p2q3 + 7pq2 Identify like terms.
There are no like terms.
Your Turn:
Identify the like terms in each polynomial.
A. 4y4 + y2 + 2 – 8y2 + 2y4
4y4 + y 2 + 2 – 8y2 + 2y 4
Identify like terms.
Like terms: 4y4 and 2y4, y2 and –8y2
B. 7n4r2 + 3n2r3 + 5n4r2 + n4r2
7n4r2 + 3n2r3 + 5n4r2 + n4r2
Like terms: 7n4r2, 5n4r2, and n4r2
Identify like terms.
Your Turn:
Identify the like terms in the polynomial.
C. 9m3n2 + 7m2n3 + pq2
9m3n2 + 7m2n3 + pq2
Identify like terms.
There are no like terms.
Example: Add or Subtract
Monomials
Simplify.
A. 4x2 + 2x2
4x2 + 2x2
Identify like terms.
6x2
Combine coefficients:
4+2=6
Example: Add or Subtract
Monomials
Simplify.
B. 3n5m4 - n5m4
3n5m4 - n5m4
2n5m4
Identify like terms.
Combine coefficients:
3 - 1 = 2.
Your Turn:
Simplify.
A. 2x3 - 5x3
2x3 - 5x3
Identify like terms.
-3x3
Combine coefficients:
2 - 5 = -3
Your Turn:
Simplify.
B. 2n5p4 + n5p4
2n5p4 + n5p4
3n5p4
Identify like terms.
Combine coefficients:
2+1=3
Definitions
A polynomial is a monomial or a sum or difference of
monomials.
Example: 3x4 + 5x2 – 7x + 1
This polynomial is the sum of the
monomials 3x4, 5x2, -7x, and 1.
The degree of a polynomial is the degree of the term
with the greatest degree.
Example: The degree of 3x4 + 5x2 – 7x + 1 is 4.
Example: Degree of a
Polynomial
Find the degree of each polynomial.
A. 11x7 + 3x3
11x7: degree 7 3x3: degree 3
The degree of the polynomial is the
greatest degree, 7.
Find the degree of
each term.
B.
:degree 3
:degree 4
Find the degree of
each term.
–5: degree 0
The degree of the polynomial is the greatest degree, 4.
Your Turn:
Find the degree of each polynomial.
a. 5x – 6
5x: degree 1
–6: degree 0
The degree of the polynomial is the
greatest degree, 1.
Find the degree of
each term.
b. x3y2 + x2y3 – x4 + 2
x3y2: degree 5
–x4: degree 4
x2y3: degree 5
2: degree 0
The degree of the polynomial is the
greatest degree, 5.
Find the degree of
each term.
Definitions
The terms of a polynomial may be written in any order.
However, polynomials that contain only one variable
are usually written in standard form.
The standard form of a polynomial that contains one
variable is written with the terms in order from greatest
degree to least degree. When written in standard form,
the coefficient of the first term is called the leading
coefficient.
Example: 3x4 + 5x2 – 7x + 1 and 3 is the leading
coefficient.
Example: Standard Form
Write the polynomial in standard form. Then give
the leading coefficient.
6x – 7x5 + 4x2 + 9
Find the degree of each term. Then arrange them in descending order:
6x – 7x5 + 4x2 + 9
Degree
1
5
2
0
–7x5 + 4x2 + 6x + 9
5
2
1
0
The standard form is –7x5 + 4x2 + 6x + 9. The leading
coefficient is –7.
Your Turn:
Write the polynomial in standard form. Then give the
leading coefficient.
16 – 4x2 + x5 + 9x3
Find the degree of each term. Then arrange them in descending order:
16 – 4x2 + x5 + 9x3
Degree
0
2
5
3
x5 + 9x3 – 4x2 + 16
5
3
2
0
The standard form is x5 + 9x3 – 4x2 + 16. The leading
coefficient is 1.
Your Turn:
Write the polynomial in standard form. Then give the
leading coefficient.
18y5 – 3y8 + 14y
Find the degree of each term. Then arrange them in descending order:
18y5 – 3y8 + 14y
Degree
5
8
1
–3y8 + 18y5 + 14y
8
5
The standard form is –3y8 + 18y5 + 14y. The leading
coefficient is –3.
1
Some polynomials have special names based on
their degree and the number of terms they have.
By Degree
Degree
Name
0
Constant
1
Linear
2
Quadratic
3
Cubic
4
Quartic
5
6 or more
Quintic
6th,7th,degree and so
on
By # of Terms
Terms
Name
1
Monomial
2
Binomial
3
Trinomial
4 or more
Polynomial
Example: Classifying
Polynomials
Classify each polynomial according to its degree and
number of terms.
A. 5n3 + 4n
Degree 3 Terms 2
5n3 + 4n is a cubic binomial.
B. 4y6 – 5y3 + 2y – 9
Degree 6 Terms 4
4y6 – 5y3 + 2y – 9 is a
C. –2x
Degree 1 Terms 1
–2x is a linear monomial.
6th-degree polynomial.
Your Turn:
Classify each polynomial according to its degree and number of
terms.
a. x3 + x2 – x + 2
Degree 3 Terms 4
x3 + x2 – x + 2 is a cubic
polynomial.
b. 6
Degree 0 Terms 1
c. –3y8 + 18y5 + 14y
Degree 8 Terms 3
6 is a constant monomial.
–3y8 + 18y5 + 14y is an 8thdegree trinomial.
Adding and Subtracting
Polynomials
Just as you can perform operations on
numbers, you can perform operations on
polynomials. To add or subtract
polynomials, combine like terms.
Example: Simplifying
Polynomials
Combine like terms.
A. 12p3 + 11p2 + 8p3
12p3 + 11p2 + 8p3
12p3 + 8p3 + 11p2
20p3 + 11p2
B. 5x2 – 6 – 3x + 8
5x2 – 6 – 3x + 8
5x2 – 3x + 8 – 6
5x2 – 3x + 2
Identify like terms.
Rearrange terms so that like
terms are together.
Combine like terms.
Identify like terms.
Rearrange terms so that like
terms are together.
Combine like terms.
Example: Simplifying
Polynomials
Combine like terms.
C. t2 + 2s2 – 4t2 – s2
t2 + 2s2 – 4t2 – s2
t2 – 4t2 + 2s2 – s2
–3t2 + s2
Identify like terms.
Rearrange terms so that like
terms are together.
Combine like terms.
D. 10m2n + 4m2n – 8m2n
10m2n + 4m2n – 8m2n
Identify like terms.
6m2n
Combine like terms.
Remember!
Like terms are constants or terms with the same
variable(s) raised to the same power(s).
Your Turn:
Combine like terms.
a. 2s2 + 3s2 + s
2s2 + 3s2 + s
5s2 + s
Identify like terms.
Combine like terms.
b. 4z4 – 8 + 16z4 + 2
4z4 – 8 + 16z4 + 2
4z4 + 16z4 – 8 + 2
20z4 – 6
Identify like terms.
Rearrange terms so that like
terms are together.
Combine like terms.
Your Turn:
Combine like terms.
c. 2x8 + 7y8 – x8 – y8
2x8
7y8
x8
y8
+
–
–
2x8 – x8 + 7y8 – y8
x8 + 6y8
Identify like terms.
Rearrange terms so that like
terms are together.
Combine like terms.
d. 9b3c2 + 5b3c2 – 13b3c2
9b3c2 + 5b3c2 – 13b3c2
b3c2
Identify like terms.
Combine like terms.
Adding Polynomials
Polynomials can be added in either
vertical or horizontal form.
In vertical form, align
the like terms and add:
5x2 + 4x + 1
+ 2x2 + 5x + 2
7x2 + 9x + 3
In horizontal form, use the
Associative and
Commutative Properties to
regroup and combine like
terms.
(5x2 + 4x + 1) + (2x2 + 5x + 2)
= (5x2 + 2x2 + 1) + (4x + 5x) + (1 + 2)
= 7x2 + 9x + 3
Example: Adding
Polynomials
Add.
A. (4m2 + 5) + (m2 – m + 6)
(4m2 + 5) + (m2 – m + 6)
Identify like terms.
(4m2 + m2) + (–m) +(5 + 6)
Group like terms
together.
Combine like terms.
5m2 – m + 11
B. (10xy + x) + (–3xy + y)
(10xy + x) + (–3xy + y)
Identify like terms.
(10xy – 3xy) + x + y
Group like terms
together.
Combine like terms.
7xy + x + y
Add.
Example: Adding
Polynomials
(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)
(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y) Identify like terms.
(6x2 + 3x2 – 8x2) + (3y – 4y – 2y)
6x2 – 4y
+ –5x2 + y
x2 – 3y
Group like terms together
within each polynomial.
Use the vertical method.
Combine like terms.
Simplify.
Your Turn:
Add.
Identify like terms.
Group like terms
together.
Combine like terms.
Your Turn:
Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a).
(5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a)
Identify like terms.
(5a3 + 7a3) + (3a2 + 12a2) + (–10a – 6a) Group like terms
together.
12a3 + 15a2 – 16a
Combine like terms.
Subtracting Polynomials
To subtract polynomials, remember that
subtracting is the same as adding the
opposite (distributing the negative). To
find the opposite of a polynomial, you
must write the opposite of each term in
the polynomial:
–(2x3 – 3x + 7)= –2x3 + 3x – 7
Example: Subtracting
Polynomials
Subtract.
(x3 + 4y) – (2x3)
(x3 + 4y) + (–2x3)
Rewrite subtraction as addition of
the opposite.
Identify like terms.
(x3 – 2x3) + 4y
Group like terms together.
–x3 + 4y
Combine like terms.
(x3 + 4y) + (–2x3)
Example: Subtracting
Polynomials
Subtract.
(7m4 – 2m2) – (5m4 – 5m2 + 8)
(7m4 – 2m2) + (–5m4 + 5m2 – 8) Rewrite subtraction as
addition of the opposite.
(7m4 – 2m2) + (–5m4 + 5m2 – 8) Identify like terms.
(7m4 – 5m4) + (–2m2 + 5m2) – 8 Group like terms together.
2m4 + 3m2 – 8
Combine like terms.
Example: Subtracting
Polynomials
Subtract.
(–10x2 – 3x + 7) – (x2 – 9)
(–10x2 – 3x + 7) + (–x2 + 9)
(–10x2 – 3x + 7) + (–x2 + 9)
–10x2 – 3x + 7
–x2 + 0x + 9
–11x2 – 3x + 16
Rewrite subtraction as
addition of the opposite.
Identify like terms.
Use the vertical method.
Write 0x as a placeholder.
Combine like terms.
Your Turn:
Subtract.
(9q2 – 3q) – (q2 – 5)
(9q2 – 3q) + (–q2 + 5)
(9q2 – 3q) + (–q2 + 5)
9q2 – 3q + 0
+ − q2 – 0q + 5
8q2 – 3q + 5
Rewrite subtraction as
addition of the opposite.
Identify like terms.
Use the vertical method.
Write 0 and 0q as
placeholders.
Combine like terms.
Your Turn:
Subtract.
(2x2 – 3x2 + 1) – (x2 + x + 1)
(2x2 – 3x2 + 1) + (–x2 – x – 1)
Rewrite subtraction as
addition of the opposite.
(2x2 – 3x2 + 1) + (–x2 – x – 1)
Identify like terms.
–x2 + 0x + 1
+ –x2 – x – 1
–2x2 – x
Use the vertical method.
Write 0x as a placeholder.
Combine like terms.
Joke Time
• How does Hitler tie his shoes?
• With little Nazis!
• What did one snowman say to the other?
• Do you smell carrots?
• There’s two fish in a tank. What did one fish say to the
other?
• You man the guns, I’ll drive!
Assignment
• 8-1 Exercises Pg. 488 - 489: #8 – 44 even