Introduction to Polynomials

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 - 3 or 5x 2
• Are these polynomials or not polynomials?

No
3/xy
yes
-2
yes
xyab
No
x 3
No
vx
Yes
(1/2)x
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 -3x4 4x2 x 1
• Term
• -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

6

• Ex2 -6x9 8x6 y4 x7 y 3xy5 - 4
• Term
• -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

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

• Opposites of Polynomials
• 2x
• Opposite is -2x
• 2) 3x4 4x2 x
• Opposite is - 3x4 4x2 - x

10
Adding and Subtracting Polynomials

• Same as combining like-term
• Add or subtract only numbers and keep the same
  variables

11

• 1) (-6x4 8x3 3x - 4) (5x4 x3 2x2 -7x)
• -6x4 5x4 8x3 x3 2x2 3x -7x -4
• -x4 - 7x3 2x2 - 4x
  -4

12

• (-6x4 8x3 3x - 4) - (5x4 x3 2x2 -7x)
• -6x4 8x3 3x - 4 - 5x4 - x3 - 2x2
  7x
• -11x4 - 9x3 - 2x2 10x -4