Transcript 5.1 Introduction to Polynomials

4.1 Introduction to
Polynomials
• Monomial: 1 term (axn with n is a nonnegative integers, a is a real number)
Ex: 3x, -3, or 4xy2z
• Binomial: 2 terms
Ex: 3x - 5, or 4xy2z + 3ab
• Trinomial: 3 terms
Ex: 4x2 + 2x - 3
• Polynomial: is a monomial or sum of
monomials
Ex: 4x3 + 4x2 - 2x - 3 or 5x + 2
• Are these polynomials or not polynomials?
3/xy
-2
xyab
| x – 3|
√x
(1/2)x
No
yes
yes
No
No
Yes
• Degree: exponents
• Degree of polynomial: highest exponent
(if the term has more than 1 variable, then
add all exponents of that term)
• Coefficient: number in front of variables
• Leading term: term of highest degree. Its
coefficient is called the leading coefficient
• Constant term: the term without variable
• Missing term: the term that has 0 as its
coefficient
• Ex:
Term:
-3x4 – 4x2 + x – 1
-3x4 , – 4x2 , x,
–1
Degree
4
2
1
0
Coefficient
-3
-4
1 -1
Degree of this polynomial
is 4
Leading term
is -3x4 and -3 is the leading coefficient
Constant term:
is -1
Missing term (s):
is x3
• Ex2:
Term:
-6x9– 8x6 y4 + x7 y + 3xy5 - 4
-6x9, – 8x6 y4 , x7 y , 3xy5 , - 4
Degree
9
10
8
6
0
Coefficient
-6
-8
1
3
-4
Degree of this polynomial
is 10
Leading term
is – 8x6 y4 and -8 is the leading coefficient
Constant term:
is -4
• Descending order: exponents decrease
from left to right
• Ascending order: exponents increase
from left to right
• When working with polynomials, we often
use Descending order
•
Arrange in descending order using power
of x
1) -6x2 – 8x6 + x8 + 3x - 4
= x8– 8x6 - 6x2 + 3x - 4
2) 5x2y2 + 4xy + 2x3y4 + 9x4
= 9x4 + 2x3y4 + 5x2y2 + 4xy
•
Opposites of Polynomials:
1) 2x
Opposite is -2x
2) 3x4 – 4x2 + x
Opposite is - 3x4 + 4x2 - x
Adding and Subtracting
Polynomials
• Same as combining like-term:
Add or subtract only numbers and keep the
same variables
1) (-6x4 – 8x3 + 3x - 4) + (5x4 + x3 + 2x2 -7x)
= -6x4 + 5x4 – 8x3 + x3 + 2x2 + 3x -7x -4
=
-x4
- 7x3 + 2x2
- 4x
-4
2) (-6x4 – 8x3 + 3x - 4) - (5x4 + x3 + 2x2 -7x)
= -6x4 – 8x3 + 3x - 4 - 5x4 - x3 - 2x2 +7x
= -11x4 - 9x3 - 2x2 +10x -4