CP Algebra 2 Unit 3A: Polynomials - State College Area School District

CP Algebra 2 Unit 3A: Polynomials

Name: _____________________________________ Period: _____

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Learning Targets

1. I can classify polynomials by degree and number of terms. Polynomials: 2. I can use polynomial functions to model real life situations and make predictions.

The Basics 3. I can identify the characteristics of a polynomial function, such as the intervals of increase/decrease, intercepts, domain/range, relative minimum/maximum, and end behavior. 4. I can write standard form polynomial equations in factored form and vice versa.

Factors and 5. I can find the zeros (or x-intercepts or solutions) of a polynomial in factored form and Zeros identify the multiplicity of each zero. 6. I can write a polynomial function from its real roots. 7. I can use long division to divide polynomials.

Dividing 8. I can use synthetic division to divide polynomials. Polynomials

9. I can use synthetic division and the Remainder Theorem to evaluate polynomials.

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Polynomials: The Basics

After this lesson and practice, I will be able to ... ? classify polynomials by degree and number of terms. (LT 1)

? use polynomial functions to model real life situations and make predictions. (LT 2)

? identify the characteristics of a polynomial function, such as the intervals of increase/decrease, intercepts, domain/range, relative minimum/maximum, and end behavior. (LT 3)

-------------------------------------------------------------------------------------------------------------------------------------------------Quadratic functions: Functions whose greatest exponent on any variable is _________

Polynomial functions: Functions whose greatest exponent on any variable is greater than _________

Classify Polynomials (LT 1)

It will be easier to talk about polynomials if we share some common vocabulary ...

Monomial ? A _________________, __________________, or ____________ of numbers and variables.

Examples:

Polynomial ? A ___________________ or ____________ of monomials.

Standard Form:

y

=

a xn n

+

an-1x n-1

+

...

+

a1x

+

a0

where

an , an-1,..., a1, a0

are the ______________ of the powers

of x, and n is a nonnegative integer.

? The exponents of the variables are given in _______________ order when written in general form.

? Each of the monomials being added is called a _________.

Examples:

Degree of a Polynomial ? The _______________ of the term with the greatest _______________. Leading Coefficient ? The ___________________ of the term with the greatest _______________.

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There are also special names that we give to polynomials based on their degree and their number of terms.

Degree Name 0 1 2 3 4 5

Example

Number Name

of Terms

Example

Example 1: Write each polynomial in standard form. Then classify each polynomial by its degree and number of terms. Finally, name the leading coefficient of each polynomial.

a.

!-7x +5x4

b.

!x2 - 4x + 3x3 + 2x

c. !4x - 6x +5

d.

!6 - 3x5

Model Real Life Situations (LT 2)

We will use Desmos to explore some characteristics of polynomials. For now, let's get an idea of the general shape of polynomials of different degrees ...

= ! - - 1

= ! + 6! + ! - 1

!

= ! + 3! + ! - - 1

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A relative minimum or maximum is a point that is the min. or max. __________________ to other nearby function values.

Example 2: Graph the equation !y = 3x3 -5x +5 in your calculator. Then determine the coordinates of all relative minimums and maximums (rounded to 3 decimal places). Here are the directions:

- Enter the equation in Y1. Press GRAPH. - If necessary, adjust the window so that you can see all relative max/min. - For finding relative maximums/minimums:

- Press 2nd CALC, 3 (minimum) or 4 (maximum) - Give the calculator a left and right bound for the maximum/minimum and then press ENTER again when the calculator asks for a guess.

Example 3: Determine the coordinates of all relative minimums and maximums (rounded to 3 decimal places).

a.

!0.5x4 - 3x2 + 3

b.

!- x3 + 6x2 - x - 1

You have already used your calculator to model data using ______________ regression and _______________ regression. Let's explore how to use other regression models to develop models for real-life data.

Example 4: a. Graph the following data in the calculator. Then determine whether a linear, quadratic, or cubic regression model best represents the data.

- Enter your data (STAT, Edit...) - Turn on Stat Plot 1 (2nd, STAT PLOT)

x 0 5 10 15 20 y 10.1 2.8 8.1 16.0 17.8

- Graph the data (ZOOM, 9)

- STAT, right, then choose your desired regression.

- Your command should look like ____Reg, Y1 (VARS, right, ENTER, ENTER)

b. Write the equation that best represents the data:

c. Predict the value of y when x is 12.

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