Introduction to Polynomials

Introduction to Polynomials

Math 9

Multiple Choice

What is a polynomial?

• one or more terms (separated by addition or subtraction) whose variables have whole number exponents

• one or more terms (separated by multiplication or division) whose variables have whole number exponents

• one or more terms (separated by addition or subtraction) whose variables have integer number exponents

• one or more terms (separated by multiplication or division) whose variables have integer number exponents

Polynomials

A polynomial is one or more terms whose variables have whole number exponents. 

 $3x^2-2x+1$  

 $12x-8$  

 $\frac{1}{2}x$  

Multiple Select

Which of the following are polynomials?

• $\frac{2}{3}x^3$

• $\frac{3}{x}+12$
  

• $5x^{-2}-9$

• $6x^7-2x^5+3x^3-x$

Terminology

• Terms: are numbers, variables, or the product of both. In the examples

   $12,\ x^2,\ 5a$  , 12 is a number,  $x^2$  is a variable, and  $5a$  is the product of a number and variable.

• Coefficients: the numerical factor of a term or number multiplied by a variable. In the example  $-5x$  , -5 is the coefficient.

Terminology cont.

• Degree: The term with the largest exponent or sum of exponents. In the example  $-9x^8+10x^{15}-1$  , 15 is the largest exponent so the degree would be 15. In the example  $7a^5b^2-12a^4b$  ,  $7a^5b^2$  has a sum of 7 and  $12a^4b$   has a sum of 5. Since 7 is larger the degree is 7.

• Constant: a constant is a number in which the variable value has no impact. The number is without a variable. Eg.  $x^2+10x-8$  has a constant term of  $-8$ , remember to take the sign in front of the constant as well.

Fill in the Blanks

In the example

 $5v^3-7$  the constant is:

Fill in the Blanks

How many terms are in the polynomial

 $3x^2+4x-8$  

Fill in the Blanks

What are the coefficients of the polynomial

 $-4n^5+6n^4-7$  , separate them with a comma

Fill in the Blanks

What is the degree of the polynomial

 $3x^8y+7xy^2-5x^5y^4$  

Classifying Polynomials 

• Polynomials can be classified based on how many terms exist in the polynomial

• Monomial: 1 term, no addition or subtraction signs.

• Monomial Examples:  $5,\ 3xy,\ -8abc,\ p$  

Classifying Polynomials Cont.

• Binomial: 2 terms, one addition or subtraction sign separating terms

• Binomial Examples: $3x-5,\ 7a^2b\ +8a,\ -6p\ -2$  

  
  

Classifying Polynomials Cont.

• Trinomial: 3 terms, two addition or subtraction signs separating terms

• Trinomial Examples: $3x^2+5x-9,\ \ 7pq\ +p^2q-10$  

Classifying Cont.

Anything above a trinomial we call a polynomial, poly meaning many

Multiple Choice

Which of the following is a trinomial

• $5x^2+12$

• $3a^2b+8$

• $9pqr$

• $5x^2-3x+17$

Ordering Polynomials

A polynomial is usually written in descending order; that is, from the largest degree to the smallest degree. It is important to note that when you rearrange terms you MUST take the sign (+/-) in FRONT of the term.

Eg.

 $3a+2a^2-4$  would become  $2a^2+3a-4$  
 $-6r^3-4r^8+2r^9$  would become  $2r^9-4r^8-6r^3$  
 $7a-9a^5-8+2a^3$  would become  $-9a^5+2a^3+7a-8$  

Multiple Select

Which of the following polynomials are in "standard form" or the correct order

• $3x^8+2x^5-9$

• $10+8p^2-6p$

• $4a^2-14$

• $5y^8+5x^7+5x^6$

• $144-x^2$