6.2 Polynomials

6.2 Polynomials

Types of polynomials

Evaluate Polynomials

Add and subtract Polynomials

What is a polynomial?

What is it made of?

• An expression with one or more terms

• contains constants, variables, and exponents

• terms can be added, subtracted, or multiplied

These are polynomials

•  $2x$  

•  $3x^2+4y^3-\ z$  
  

• 5

What is a polynomial?

What can it not have?

• no fractional exponents

• No variables in the denominator

NOT polynomials

•  $x^{\frac{1}{2}}$  

•  $\frac{2}{x+2}$  

Poll

Is this a polynomials?

 $5x^7$  

• Yes

• No

• I am unsure 

Poll

Is this a polynomial?

 $x^{\frac{2}{3}}\ +\ y^{\frac{4}{5}}$  

• Yes

• No

• I am unsure

Types of polynomials

There are 4 types to consider.

You can tell them apart based on the number of terms they have.

(Terms are separated by + and - signs)

Monomial

• One term

• Can be a constant, a variable, or a combination

Examples of monomials

•  $y$  

•  $8$  

•  $3x^4$  

•  $5x^3y^2z^8$  

Poll

Is this a monomial?

 $2x+7$  

• Yes

• No

• I am unsure

Binomials

• 2 terms

• Terms are separated by a plus or minus sign

Examples of Binomials

•  $x+1$  

•  $3x^3-\ 4y$  

•  $7x^5-3x^2$  

•  $4x+9$  

Multiple Choice

Classify.

 $2x^8$  

• monomial

• binomial

• neither

Multiple Choice

Classify

 $x^2+9$  

• monomial

• binomial

• neither

Multiple Choice

Classify

 $x^2-3x+6$  

• monomial

• binomial

• neither

Trinomials

• 3 terms

• Terms separated by plus or minus signs

Examples of Trinomials

•  $x^2+3x-9$  

•  $2x^2+4x+3$  

•  $8a^2-7a-6$  

Multiple Choice

Classify

 $8$  

• monomial

• binomial

• trinomial

• neither

Multiple Choice

Classify.

 $x^2-7x+9$  

• monomial

• binomial

• trinomial

Multiple Choice

A binomial has how many terms?

• one

• two

• three

• infinitely many

Polynomials

• monomials,binomials, and trinomials are specific types of polynomials

• If there are 4 or more terms, we just call it a polynomial

Evaluate Polynomials

* Plug in the value given for the variable(s)

* Simplify using PEMDAS

If x= 3 find

 $2x^2-3x+4$  

• replace each x with a 3

• Simplify

If x = -2 find

 $-3x^3+2x^2-x+5$  

• Replace each x with -2

• Simplify

Multiple Choice

If x= 4 find

 $2x^2-3x\ +1$  

• 5

• 53

• 21

• 245

Multiple Choice

If x= - 3 find

 $-2x^2+4x\ -8$  

• -8

• -2

• -56

• -38

Add and Subtract Polynomials

• Combine like terms

• Like terms have the same variables and same exponents

Poll

Are these like terms?

 $2x^3\ and\ 9x^3$  

• yes

• no

• I am unsure

Poll

Are these like terms?

 $x^2y^2\ and\ x^2y^{19}$  

• yes

• no

• I am unsure

Add the polynomials

Find the terms that are alike

Simplify by combining like terms

Subtract Polynomials

• Distribute negative sign

• Combine like terms

Multiple Choice

Simplify

 $\left(4x-3\right)\ +\ \left(8x+7\right)$  

•  $4x+4$  

•  $12x+10$  

•  $4x+10$  

Multiple Choice

Simplify

 $\left(2x^2-3x+4\right)\ -\ \left(-4x^2+6x-8\right)$  

•  $-2x^2+3x-4$  

•  $6x^2+3x\ -\ 4$  

•  $6x^2-9x+12$