Polynomial Functions

5.1 - Polynomial Functions

Objective

To classify polynomials
To graph polynomial functions and describe end behavior.

Definitions

• Monomial: a real number, a variable, or a product of real numbers and variables with whole number exponents

• Polynomial: a monomial or a sum or difference of monomials

• Polynomials have no variables in denominators or exponents, no roots or absolute values of variables, and all variables have whole number exponents.

Classifying Polynomials

• The degree of a polynomial is the degree of the term with the greatest degree (highest exponent)

• The leading coefficient is the coefficient of the first term

A polynomial is written in standard form when:

• The terms are written in descending order by degree

• All like terms have been combined

A polynomial can be clasified by its number of terms

A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial.

Multiple Choice

Classify the polynomial:

$$14x^3+x^2-x+2$$  

• Quadratic trinomial

• Triad Quintomino

• Cubic with four terms

• Fourth degree polynomial

Multiple Choice

Write the polynomial in standard form:

$$7x^3-11x+x^5-2$$  

• $$-11x+7x^3-2+x^5$$  

• $$x^5+7x^3-11x-2$$  

• $$7x^3-11x+x^5-2$$  

• $$18x^5-2$$  

Multiple Select

Check all the expressions that are polynomials:

• $$3x^4$$

• $$3^x$$

• $$\frac{1}{2}a^7$$

• $$\frac{1}{2}\sqrt{x}$$

• $$\frac{8}{5y^2}$$

Open Ended

Why is

$$\frac{6}{x^3}$$   not a polynomial?

The degree of a polynomial function affects the shape of the graph and determines the number of turning points (the point or points where a graph changes direction).

If a turning point is higher than any nearby point, its called a relative maximum.

If a turning point is lower than any nearby point, its called a relative minimum.

The maximum and minimum values are called extrema.

The turning points define where a function is increasing or decreasing.

End behavior

End behavior

End behavior

Drag and Drop

The function f(x) = -x³ - x² + 8x + 1 has degree ​

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with a leading coefficient of ​

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and has end behavior of ​

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.

Drag these tiles and drop them in the correct blank above

3

-1

up -down

2

8

down-up

up-up

down-down

4

Good work!

Please complete the last section of the notes handout.

Poll

Select the image that most closely reflects your understanding of characteristics of polynomial functions: