Introduction to Polynomial

8.1 Addition and Subtraction of Polynomials

Algebra 1

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.

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

• $$3x^8+2x^5-9$$

• $$10+8p^2-6p$$

• $$4a^2-14$$

• $$5y^8+5x^7+5x^6$$

• $$144-x^2$$

Multiple Choice

Find the sum.
(2x² + 5x - 7) + ( 3 - 4x² + 6x)

• 2x² + 3x +1
• -2x² - 11x -4
• 2x² + 5x -7
• -2x² + 11x -4

Multiple Choice

Find the sum.

(3x² + x³ - 5) + (4x² - 4x + 2x³ + 8)

• 3x³ + 7x² - 4x + 3

• 4x² + 3x³ - x + 3

• -x⁴ + 2x³ - x - 4

• 4x³ + 3x² - 3x + 1

Multiple Choice

(3h - 4) - (8h - 2) =

• -11h - 2
• - 5h + 2
• - 5h - 2
• - 5h - 6

Multiple Choice

(5x³- x² + 4) - (3x³ - 2x² - 3)

• -2x³ - 3x² + 7

• -2x³ + x² + 1

• 2x³ + x² + 7

• 4x³ - 3x² + 7