Section P4
Polynomials
How We Describe Polynomials
The Degree of ax n
If a  0, the degree of ax n is n. The degree of a
nonzero constant is 0. The constant 0 has no
defined degree.
Adding and Subtracting
Polynomials
Combine Like Terms
Example
Perform the indicated operations and simplify:
3
2
3
2

6
x

2
x

8

13
x

4
x
 4 x  14 

 
Example
Perform the indicated operations and simplify:
9x
3
 2 x  x  9    x  5 x  8 x  10 
2
3
2
Multiplying Polynomials
Multiplying by a monomial
Example
Find each product:
5 x  2 x  5 x  9 x  14 
2
3
2
Multiplying Polynomials When Neither is a Monomial
Multipying each term of one polynomial by each term
of the other polynomial. Then combine like terms.
Example
Find each product:
 4 x  1  x
2
 10 x  16 
The Product of Two Binomials:
FOIL
Multiplying Two Binomials
using the Distributive Property
Example
Find each product:
 7 x  63x  8
Example
Find each product:
 9 x  28x  9
Multiplying the Sum and
Difference of Two Terms
Example
Find the product:
 7 x  4  7 x  4 
Example
Find the product:
2
2
8
a

3
8
a

  3
The Square of a Binomial
 First 
2
+ 2  product +  Last  =Product
2
of terms
2
x

4

x


2
 2 x  5   2x 
2
2
+
2x 4
+
2  2x  -5  +
= x 2  8 x  16
+ 42
 -5 
2
= 4x 2  20 x  25
Example
Find each product:
 x  4
2
Example
Find each product:
 2x  9
2
Special Products
Polynomials in Several
Variables
A polynomial in two variables, x and y, contains
the sum of one or more monomials in the form
axnym. The constant a is the coefficient. The
exponents n and m represent whole numbers.
The degree of a polynomial in two variables is the
highest degree of all its terms.
Example
Perform the indicated operations:
3
2
3
2
x
y

4
x

18
xy

19

6
x

  y  7 xy  14 
2
2
5
x
y

9
xy

7

19
x

  y  4 xy  71
Example
Find the product:
 3xy  7 8 xy  9 
Example
Find the product:
4x  9 y 
2
Perform the indicated operations.
3
2
2
5
x
y

x
y

xy

7

9
x

  y  18xy  45
3

4
x
y  17 xy  52
(a)
3
2

4
x
y

19
x
y  xy  52
(b)
(c) 5 x 3 y  8 x 2 y  17 xy  38
(d) 5 x 3 y  8 x 2 y  19 xy  38
Find the product.
8x  9  7 x
2
 x  8
(a)
56 x3  55 x 2  9 x  72
(b)
56 x3  55 x 2  73x  72
(c)
56 x3  8 x 2  8 x  72
(d)
56 x3  x  136