POLYNOMIALS WHAT IS A POLYNOMIAL An algebraic expression

POLYNOMIALS

WHAT IS A POLYNOMIAL? An algebraic expression that contains more than two terms Polynomial literally means poly – (meaning many) and nomial - (meaning terms). Or in this case, many terms.

DEFINITIONS Variable: a symbol for a number we don’t know yet. In math, we will usually represent this using ‘x’ or ‘y’ Term: a single number or variable, or a combination of both. Algebraic Expression: a mathematical phrase that contains terms and operations. Constant: a term that is a number. It is not changing. Exponent: Like the 2 in y². But for the purposes of polynomials, they can only be 0, 1, 2, 3, etc

EXAMPLES OF POLYNOMIALS 2 x + 3 4 xᶾ – 3 x + 7 9 y⁸ + 14 y⁴ – 3 z

TYPES OF ALGEBRAIC EXPRESSIONS Monomial Binomial Trinomial Polynomial

MONOMIAL Containing only one term Examples: x, y, 3, 4 rᶾ, 7 m⁸, 2 xyᶾ

BINOMIAL Containing two terms Examples: 2 x + 3, 9 y – 1, r² + 5, x²v⁴ + c Where do we see binomials?

TRINOMIAL Containing three terms Examples: 2 xy + 4 z – t, ax² + bx +c, 3 + 4 mn – 7 oᶾ Where do we see trinomials?

POLYNOMIAL Containing more than three terms We will use the term polynomial to classify all expressions with two or more terms as stated previously More specifically, we will use it for four or more terms

THINK BICYCLES!!!!

We can combine polynomials using addition, subtraction, multiplication, and division Whenever we combine polynomials, we get back polynomials. This is what makes them so special and easy to work with! NOTE: We cannot divide by a variable in a polynomial. (So 2/x is NOT a polynomial)

THESE ARE POLYNOMIALS • 3 x • x - 2 • -6 y 2 - (7/9)x • 3 xyz + 3 xy 2 z - 0. 1 xz - 200 y + 0. 5 • 512 v 5+ 99 w 5 • 5

THESE ARE NOT POLYNOMIALS • 3 xy-2 is not, because the exponent is "-2" (exponents can only be 0, 1, 2, . . . ) • 2/(x+2) is not, because dividing by a variable is not allowed • 1/x is not either • √x is not, because the exponent is "½"

BUT THESE ARE x/2 is allowed, because you can divide by a constant 3 x/8 for the same reason √ 2 is allowed, because it is a constant (= 1. 4142. . . etc)

DEGREE We can classify polynomials by degree. This is the highest exponent in a polynomial. Example: 4 xᶾ -2 x +1 has degree 3

What is the degree of x²y²? What is the degree of m⁴n? What is the degree of abᶾd⁵?

Example: What is the degree of this polynomial? 4 z 3 + 5 y 2 z 2 + 2 yz Checking each term: 4 z 3 has a degree of 3 (z has an exponent of 3) 5 y 2 z 2 has a degree of 4 (y has an exponent of 2, z has 2, and 2+2=4) 2 yz has a degree of 2 (y has an exponent of 1, z has 1, and 1+1=2) The largest degree of those is 4, so the polynomial has a degree of 4

STANDARD FORM When writing a polynomial in standard form, we put the terms with the highest degree first 3 x² - 2 + 7 x⁸ - 5 xᶾ would be written as 7 x⁸ - 5 xᶾ + 3 x² - 2