polynomials. what is a polynomial? an algebraic expression that contains more than two terms...

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

4xᶾ – 3x + 7

9y⁸ + 14y⁴ – 3z

TYPES OF ALGEBRAIC EXPRESSIONS Monomial

Binomial

Trinomial

Polynomial

MONOMIAL

Containing only one term

Examples: x, y, 3, 4rᶾ, 7m⁸, 2xyᶾ

BINOMIAL

Containing two terms

Examples: 2x + 3, 9y – 1, r² + 5, x²v⁴ + c

Where do we see binomials?

TRINOMIAL

Containing three terms

Examples: 2xy + 4z – t, ax² + bx +c, 3 + 4mn – 7oᶾ

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 - 2

•-6y2 - (7/9)x

•3xyz + 3xy2z - 0.1xz - 200y + 0.5

•512v5+ 99w5

•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 (= 1.4142...etc)

DEGREE

We can classify polynomials by degree.

This is the highest exponent in a polynomial.

Example: 4xᶾ -2x +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?

4z3 + 5y2z2 + 2yz

Checking each term:

4z3 has a degree of 3 (z has an exponent of 3)

5y2z2 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 - 2 + 7x⁸ - 5xᶾ would be written as 7x⁸ - 5xᶾ + 3x² - 2

LIKE TERMS

Like terms are terms who have the same variable AND exponent

3x – 5x

2yz + 8yz

7z – 9z

3x² + 2x²

5rs⁴ - 3rs⁴

2mn⁸ + 5mn⁸ - 11mn⁸

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

5rs⁴ - 3rs⁴

2mn⁸ + 5mn⁸ - 11mn⁸

xy + 3xy – 9xy + 4xy – 2xy

If the terms are unlike terms, we cannot collect them.

x² + xᶾ x⁵

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

-3yzᶾ -6x⁴z² + yz + 1 and -4yzᶾ +2x⁴z² + yz - 5

SUBTRACTION

6b²c + y + b and 6b²c +2y² - b

SIMPLIFY THE FOLLOWING EXPRESSIONS 8x + 2y – z + 2x – 6y + 4

3st² – 5s – 6st² +7s + tᶾ

YOUR TURN!!!!