Polynomial Introduction

Introduction to Polynomials

Objective: Identify the degree and lead coefficient of a given polynomial.

​TERM

- a number, a variable, or the product of a number and variable(s)

​EXAMPLES

​​poly- "many" -nomial "term" Term(s) whose exponents are whole numbers, and are separated by a "+" or a "-"

​POLYNOMIAL

​​x + 3

​​x² - 4x - 9

​9​x³ + 5x² - 3x + 12

​How many terms does each polynomial have?

​Degree of a term - is the sum of the exponents of each variable in a term.

​Degree of a polynomial - is the greatest value of the sum of the exponents of each variable in a term.  

​DEGREE OF A POLYNOMIAL

​EXAMPLES

​3x² + 9x - 4

​x² - 8x⁵ + 7

​y⁵z⁴ + w²x³ + 2xy - 10

​the degree is ___

​the degree is ___

​the degree is ___

the highest degree of its monomials with non-zero coefficients.

​LEAD COEFFICIENT

the coefficient of the first term when the polynomial is written in standard form.

​EXAMPLE

What is the lead coefficient?

​-9x³ + 3x - 6x⁴ + 22

Math Response

What is the lead coefficient of the polynomial?

$$-4a^3+19a^4-a+12a^6+103$$  

Type answer here

Correct Answer: 12

Deg^°

Rad

Math Response

What is the degree of the polynomial?

$$a^3b^3+c^4d-x^3y^2$$  

Type answer here

Correct Answer: 6

Deg^°

Rad

​NAMING A POLYNOMIAL

​by the number its degree

​DEGREE	NAME​	​EXAMPLE
0​	CONSTANT​	12​
1​	LINEAR​	​3x - 12
2​	QUADRATIC​	​
3​	CUBIC​	​
4​	4th DEGREE POLYNOMIAL	​

​x² + 4x - 8

4x³ - 5​x² + 4x

​9x⁴ - 2x² - 6

Multiple Choice

Name the polynomial by its degree.

$$3x^2+7x-8$$  

• constant

• linear

• quadratic

• cubic

Math Response

How many terms does the polynomial have?

$$2ab^2-4c+6d^3-12ef\ +\ 101$$  

Type answer here

Correct Answer: 5

Deg^°

Rad

​STANDARD FORM of a POLYNOMIAL

all of the terms are in order from the highest exponent (degree) to the lowest exponent (degree).

​EXAMPLE

Write in STANDARD FORM

​5y² + w⁴ + w⁷ - q³ + 2y - 10

Reorder

Rewrite $$2x^2+9-5x^3+3x^4-8x$$ in standard form.

• $$+3x^4$$

• $$-5x^3$$

• $$+2x^2$$

• $$-8x$$

• $$9$$

1

2

3

4

5

• 1
• 2
• 3
• 4
• 5

​NAMING A POLYNOMIAL

​by the number of terms it has

​Number of Terms	​Name	​Example
​1	MONOMIAL​	​4x
2​	BINOMIAL​	​7x - 8
3​	TRINOMIAL​	​2x - 7y + 12

Multiple Select

Chose all the descriptions that describe the polynomial.

$$-2x^2-7$$  

• trinomial

• linear

• quadratic

• binomial

• monomial

Adding and Subtracting Polynomials

• Step 1: Distribute the "1s" to eliminate the parentheses.

• Step 2: Combine like terms

Ex 1 (-7x²-180x+5800) + (21x²-140x+1900)

• Since the number in front of both sets of parentheses is 1, this eliminates the parentheses

• -7.1x²-180x+5800+21x²-140x+1900

• Combine like terms

• 14x²-320x+7700

Ex 2 (x³-3x²+5x) - (7x³+5x²-12)

• Step 1: Notice there is a -1 in front of the 2nd set of parentheses. This changes the signs of the second polynomial.

  x³-3x²+5x -7x³-5x²+12

• Step 2: Combine Like Terms

  -6x³-8x²+5x+12

Multiple Choice

Simplify (-4m³-m+9)-(4m²+m-12)

• -4m³-4m²-2m-21

• -4m³-4m²-2m+21

• -8m³-2m+21

• -4m³+4m²+2m+21

Multiple Choice

7x (3x² + 2x - 3)

• 21x² + 14x -21

• 10x³ + 9x² - 10x

• 21x³ + 14x² - 21x

Let's Learn about Special Cases

Multiplying Polynomials using Distributive property