6/23/2015
A polynomial is an algebraic
expression that includes
addition, subtraction,
multiplication, and whole
number exponents, such as:
4x3 – 3x2 + 7x + 5
A polynomial can not include
• roots or fractional
exponents
• fractions with the variable
on the bottom (or negative
exponents)
• variables in the exponent
“Polynomial” means “many
parts”.
• The parts of a polynomial
are called terms.
• Each + or – starts a new
term.
• 4x3 – 3x2 + 7x + 5
has four terms.
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How many terms are in each
of these polynomials?
• x2 – 2x + 3
How many terms are in each
of these polynomials?
• x2 – 2x + 3
3
•
4n5 + 2n4 + 3n3 – 2n2
•
4n5 + 2n4 + 3n3 – 2n2
4
•
5x2y5 + 2x3y6
•
5x2y5 + 2x3y6
2
In the polynomial
4x3
•
•
+
2y2
– 5z + 2
How many terms are there?
What is the coefficient on the 1st
term?
What is the coefficient on the 3rd
term?
What is the base on the 2nd term?
Which term is a constant?
•
•
•
In the polynomial
4x3 + 2y2 – 5z + 2
•
•
How many terms are there?
4
What is the coefficient on the 1st
term?
4
What is the coefficient on the 3rd
term?
-5
nd
What is the base on the 2 term? y
Which term is a constant?
4th
•
•
•
We often classify polynomials
by the number of terms they
have. In particular …
We often classify polynomials
by the number of terms they
have. In particular …
•
•
•
•
•
•
Monomial
Binomial
Trinomial
Monomial =
Binomial =
Trinomial =
1 term
2 terms
3 terms
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Your book will sometimes ask
you to write polynomials in
standard form. This just
means re-arranging the terms
so the exponents count down
from left to right.
You can have more than one
variable in a polynomial.
For instance 5x2y4 + 2xy3 is a
binomial where each term has
2 variables.
The degree of a term tells
how big that term is.
The degree is the sum of the
exponents in a term.
What is the degree of each
term of this polynomial?
5x2y2 – 2x5 + 4x7y3 – 7x + 5
In 5x2y4 + 2xy3, the degree of
the 1st term is 6 and the
degree of the 2nd term is 4.
What is the degree of each
term of this polynomial?
5x2y2
4
–
2x5
5
+
4x7y3
– 7x + 5
10
1 0
The degree of a whole
polynomial is the largest of the
degrees of its terms.
So, the degree of
5x2y4 + 2xy3 is 6.
The degree of
5x2y2 – 2x5 + 4x7y3 – 7x + 5
is 10.
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What is the degree of
5n5p3 + 2n2p7 – 4n4p3 + 5np6?
Special types of polynomials:
• Quadratic
•
One variable
Degree = 2
Cubic
One variable
Degree = 3
What is the degree of
5n5p3 + 2n2p7 – 4n4p3 + 5np6?
8
9
7
7
The overall degree is 9.
It’s easy to add or subtract
polynomials.
•
•
Just combine like terms.
When you do that, you
add the coefficients, and
leave the variables and
coefficients alone.
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(5x2 + 3x – 2) + (3x2 – 7x + 8)
=
8x2
– 4x + 6
The variables and exponents
act like a label for the terms,
which is why they don’t
change when you add or
subtract.
Simplify:
Simplify:
(6x2y4 + 2x3y + 3x2y2)
+ (5x3y – 2x2y4 + 2x2y2)
(6x2y4 + 2x3y + 3x2y2)
+ (5x3y – 2x2y4 + 2x2y2)
= 4x2y4 + 7x3y + 5x2y2
Simplify:
Simplify:
(y3 – 2y2 + 3y + 1) + (3y3 + 2y2 – 5y + 2)
(y3 – 2y2 + 3y + 1) + (3y3 + 2y2 – 5y + 2)
4y3 – 2y + 3
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Simplify:
Simplify:
(7n4 + 8n3 – 5n2) – (2n4 + 3n3 + n2)
(7n4 + 8n3 – 5n2) – (2n4 + 3n3 + n2)
5n4 + 5n3 – 6n2
Simplify:
Simplify:
(5x2 – 3x + 5) – (2x2 + 4x – 1)
(5x2 – 3x + 5) – (2x2 + 4x – 1)
3x2 – 7x + 6
Simplify:
Simplify:
(7x2 + 2y2) – (4x2 + 2xy – 3y2)
(7x2 + 2y2) – (4x2 + 2xy – 3y2)
3x2 – 2xy + 5y2
6