Transcript File

7-5 Polynomials
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.
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7-5 Polynomials
Example : Finding the Degree of a Monomial
Find the degree of each monomial.
A. 4p4q3
The degree is 7.
B. 7ed
The degree is 2.
C. 3
The degree is 0.
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Add the exponents of the
variables: 4 + 3 = 7.
Add the exponents of the
variables: 1+ 1 = 2.
Add the exponents of the
variables: 0 = 0.
7-5 Polynomials
A polynomial is a monomial or a sum or
difference of monomials.
The degree of a polynomial is the
degree of the term with the greatest
degree.
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7-5 Polynomials
Example : Finding the 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
–5: degree 0
:degree 4
Find the degree of
each term.
The degree of the polynomial is the greatest degree, 4.
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7-5 Polynomials
Check It Out!
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.
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Find the degree of
each term.
7-5 Polynomials
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.
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7-5 Polynomials
Example : Writing Polynomials in 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.
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7-5 Polynomials
Check It Out!
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.
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7-5 Polynomials
Some polynomials have special names based on
their degree and the number of terms they have.
Degree
Name
Terms
Name
0
Constant
1
Monomial
1
Linear
2
Binomial
2
Quadratic
Trinomial
3
4
Cubic
Quartic
3
4 or
more
5
Quintic
6 or more
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6th,7th,degree
and so on
Polynomial
7-5 Polynomials
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.
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6th-degree polynomial.
7-5 Polynomials
Check It Out!
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 polymial.
b. 6
Degree 0 Terms 1
6 is a constant monomial.
c. –3y8 + 18y5 + 14y
Degree 8 Terms 3
–3y8 + 18y5 + 14y is an
8th-degree trinomial.
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7-5 Polynomials
Just as you can perform operations on
numbers, you can perform operations on
polynomials. To add or subtract
polynomials, combine like terms.
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7-5 Polynomials
Example: Adding and Subtracting Monomials
Add or Subtract..
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
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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.
7-5 Polynomials
Remember!
Like terms are constants or terms with the same
variable(s) raised to the same power(s). To
review combining like terms, see lesson 1-7.
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7-5 Polynomials
Check It Out!
Add or subtract.
a. 2x8 + 7y8 – x8 – y8
2x8
7y8
+
–
–
2x8 – x8 + 7y8 – y8
x8 + 6y8
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x8
y8
Identify like terms.
Rearrange terms so that
like terms are together.
Combine like terms.
7-5 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
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7-5 Polynomials
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
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7-5 Polynomials
To subtract polynomials, remember that
subtracting is the same as adding the
opposite. To find the opposite of a
polynomial, you must write the opposite
of each term in the polynomial:
–(2x3 – 3x + 7)= –2x3 + 3x – 7
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7-5 Polynomials
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
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Combine like terms.
7-5 Polynomials
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
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Rewrite subtraction as
addition of the opposite.
Identify like terms.
Use the vertical method.
Write 0x as a placeholder.
Combine like terms.
7-5 Polynomials
Check It Out!
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
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Use the vertical method.
Write 0x as a placeholder.
Combine like terms.
7-5 Polynomials
Lesson Quiz: Part I
Find the degree of each polynomial.
1. 7a3b2 – 2a4 + 4b – 15
2. 25x2 – 3x4
5
4
Write each polynomial in standard form. Then
give the leading coefficient.
3. 24g3 + 10 + 7g5 – g2 7g5 + 24g3 – g2 + 10; 7
4. 14 – x4 + 3x2
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–x4 + 3x2 + 14; –1
7-5 Polynomials
Lesson Quiz: Part II
Classify each polynomial according to its
degree and number of terms.
5. 18x2 – 12x + 5
6. 2x4 – 1
quadratic trinomial
quartic binomial
Add or subtract.
7. 7m2 + 3m + 4m2
11m2 + 3m
8. (r2 + s2) – (5r2 + 4s2)
(–4r2 – 3s2)
9. (10pq + 3p) + (2pq – 5p + 6pq)
10. (14d2 + 1) + (6d2 – 2d - 8)
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18pq – 2p
20d2 – 2d – 7
7-5 Polynomials
Lesson Quiz: Part III
Add or subtract.
11. 7m2 + 3m + 4m2 11m2 + 3m
12. (r2 + s2) – (5r2 + 4s2) (–4r2 – 3s2)
13. (10pq + 3p) + (2pq – 5p + 6pq) 18pq – 2p
14. (14d2 + 1) + (6d2 – 2d - 8)
15. (2ab + 14b) – (–5ab + 4b)
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20d2 – 2d – 7
7ab + 10b