1. 4.1 Introduction to Polynomials

2. Monomial: 1 term (axn with n is a non- negative 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

3. Polynomial: is a monomial or sum of monomials
Ex: 4x3 + 4x2 - 2x or 5x + 2
Are these polynomials or not polynomials?
3/xy No -2
yes
xyab
yes
| x – 3| No √x No
(1/2)x
Yes

4. 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

5. Ex: x4 – 4x2 + x – 1
Terms:
-3x4 , – 4x2 , x, – 1
Degree
Coefficient
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

6. Ex2: x9– 8x6 y4 + x7 y + 3xy5 - 4
Terms:
-6x9, – 8x6 y4 , x7 y , 3xy5 , - 4
Degree
Coefficient
Degree of this polynomial is 10
Leading term is – 8x6 y4 and -8 is the leading coefficient
Constant term: is -4

7. 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

8. Arrange in descending order using power of x
-6x2 – 8x6 + x8 + 3x - 4
= x8– 8x6 - 6x2 + 3x - 4
5x2y2 + 4xy + 2x3y4 + 9x4
= 9x4 + 2x3y4 + 5x2y2 + 4xy

9. Adding and Subtracting Polynomials
Same as combining like-term:
Add or subtract only numbers and keep the same variables

10. 1) (-6x4 – 8x3 + 3x - 4) + (5x4 + x3 + 2x2 -7x) = -6x4 + 5x4 – 8x3 + x3 + 2x2 + 3x -7x -4 = -x4 - 7x3 + 2x2 - 4x -4

11. (-6x4 – 8x3 + 3x - 4) - (5x4 + x3 + 2x2 -7x)
= -6x4 – 8x3 + 3x x4 - x x2 +7x
= -11x4 - 9x3 - 2x2 +10x -4

12. Multiplying Polynomials
Monomial times a polynomial – Distribute!
Binomial times a binomial – Distribute! (FOIL)
Binomial times a trinomial – Distribute!