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 2x + 3 4x – 3x + 7 9y ⁸ + 14y ⁴ – 3z

TYPES OF ALGEBRAIC EXPRESSIONS Monomial Binomial Trinomial Polynomial

MONOMIAL Containing only one term Examples: x, y, 3, 4r, 7m ⁸, 2xy ᶾ

BINOMIAL Containing two terms Examples: 2x + 3, 9y – 1, r² + 5, x²v ⁴ + c Where do we see binomials?

TRINOMIAL Containing three terms Examples: 2xy + 4z – t, ax² + bx +c, 3 + 4mn – 7o 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 3x x y 2 - ( 7 / 9 )x 3xyz + 3xy 2 z - 0.1xz - 200y v w 5 5

THESE ARE NOT POLYNOMIALS 3xy -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 3x/8 for the same reason √2 is allowed, because it is a constant (= etc)

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

What is the degree of x²y²? What is the degree of m ⁴n? What is the degree of abᶾd⁵?

Example: What is the degree of this polynomial? 4z 3 + 5y 2 z 2 + 2yz Checking each term: 4z 3 has a degree of 3 (z has an exponent of 3) 5y 2 z 2 has a degree of 4 (y has an exponent of 2, z has 2, and 2+2=4) 2yz 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 3x² x ⁸ - 5x ᶾ would be written as 7x ⁸ - 5x ᶾ + 3x² - 2

LIKE TERMS Like terms are terms who have the same variable AND exponent 3x – 5x 2yz + 8yz 7z – 9z

3x² + 2x² 5rs ⁴ - 3rs ⁴ 2mn ⁸ + 5mn ⁸ - 11mn ⁸ xy + 3xy – 9xy + 4xy – 2xy

In all of the previous examples, we can collect the ‘like terms’ to reduce all of our expressions Let’s try!

3x – 5x 2yz + 8yz 7z – 9z

3x² + 2x² 5rs ⁴ - 3rs ⁴ 2mn ⁸ + 5mn ⁸ - 11mn ⁸ xy + 3xy – 9xy + 4xy – 2xy

TODAY WE WILL LOOK AT ADDING AND SUBTRACTING POLYNOMIALS

ADDITION 3x + 2 and 4x + 1

ADDITION 4x – 8yz and 7x – 3yz

ADDITION -x² + 3 and 2x + 1

ADDITION 4xy² - 3x + 8 and -7xy² +5x

SUBTRACTION x + 7st and x – 9st

SUBTRACTION 3x + 4x²y and -5x – 6x²y

SUBTRACTION -3yz -6x ⁴ z² + yz + 1 and -4yz +2x ⁴ z² + yz - 5

SUBTRACTION 6b²c + y + b and 6b²c +2y² - b

SIMPLIFY THE FOLLOWING EXPRESSIONS 8x + 2y – z + 2x – 6y + 4 3st² – 5s – 6st² +7s + t

YOUR TURN!!!!