Introduction to Polynomials

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

Polynomial: is a monomial or sum of monomials Ex: 4x3 + 4x2 - 2x - 3 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

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: -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

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

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 -6x2 – 8x6 + x8 + 3x - 4 = x8– 8x6 - 6x2 + 3x - 4 5x2y2 + 4xy + 2x3y4 + 9x4 = 9x4 + 2x3y4 + 5x2y2 + 4xy

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