39 Algebra 2 Honors

Chapter 2 Notes: Polynomials and Polynomial Functions

Section 2.1: Use Properties of Exponents

Evaluate each expression

(34)2

(5

8)

3

(−2)−3(−2)9

(𝑎2

𝑏−3)

3

(−𝑦2)5𝑦2𝑦−12 𝑟𝑠

(𝑟𝑠−1)3

More Challenge:

(𝑎2𝑏

𝑎−3𝑏2)−1

2𝑚𝑛2(3𝑚2𝑛)2

12𝑚3𝑛4

40

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Section: 2.3 Add, Subtract, and Multiply Polynomials

Examples:

(𝒙𝟒 + 𝟐𝒙𝟑 + 𝟖) + (𝟐𝒙𝟒 − 𝟗) (𝒙 + 𝟐)(𝟓𝒙𝟐 + 𝟑𝒙 − 𝟏)

(𝟗𝒙𝟒 − 𝟏𝟐𝒙𝟑 + 𝒙𝟐 − 𝟖) − (𝟑𝒙𝟒 − 𝟏𝟐𝒙𝟑 − 𝒙) (𝒙 − 𝟐)(𝒙 − 𝟏)(𝒙 + 𝟑)

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Special Product Patterns

1) Sum and Difference 2 2( )( )a b a b a b

2) Square of a Binomial

2 2 2( ) 2a b a ab b

2 2 2( ) 2a b a ab b

3) Cube of a Binomial 3 3 2 2 3( ) 3 3a b a a b ab b 3 3 2 2 3( ) 3 3a b a a b ab b

Examples:

(3 2)(3 2)x x 2(5 2)a

3( 2)ab

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Section 2.2: Evaluate and Graph Polynomial Functions

Polynomial Function: A function where a variable x is raised to a nonnegative integer power.

The domain of any polynomial is the set of all real numbers.

n is the degree of a polynomial is the highest power of x in the polynomial. The coefficient of the highest

degree term 𝑎𝑛 is called the leading coefficient. The term in a polynomial with no variable 𝑎0 is called the

constant term and is a coefficient of itself.

Example:

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What are examples of functions that are not polynomial functions?

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End Behavior: There are four scenarios:

Applying these principles to polynomials in standard form 1 2 2

1 2 2 1 0( ) ...n n n

n n nf x a x a x a x a x a x a

.

The end behavior of the polynomial depends on the degree, n, of the polynomial.

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Section: 2.4 Factor and Solve Polynomial Equations

Recall how to Factor Quadratic Equations…

Factor Polynomial in Quadratic Form:

𝟐𝟓𝒙𝟒 − 𝟑𝟔 𝒂𝟐𝒃𝟐 − 𝟖𝒂𝒃𝟑 + 𝟏𝟔𝒃𝟒

Factor by grouping:

𝑥2𝑦2 − 3𝑥2 − 4𝑦2 + 12

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Solving Polynomial Equations by Factoring:

Solve: 2𝑥5 + 24𝑥 = 14𝑥3

You are building a rectangular bin to hold mulch for your garden. The bin will hold 162 𝑓𝑡3 of

mulch. The dimensions of the bin are 𝑥 𝑓𝑡. 𝑏𝑦 5𝑥 − 6 𝑓𝑡. 𝑏𝑦 5𝑥 − 9 𝑓𝑡. How tall will the bin

be?

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Graphing Polynomials in Factored Form

Sketch the graph:

𝑔(𝑥) = −(𝑥 − 1)3(𝑥 + 2)(𝑥 − 3) 𝑓(𝑥) = (𝑥 + 3)2(𝑥 + 1)(𝑥 − 2)3

𝑓(𝑥) = −𝑥5 + 4𝑥3

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2.5 Apply the Remainder and Factor Theorems

Dividing Polynomials

When you divide a polynomial ( )f x by a divisor ( )d x , you get a quotient polynomial ( )q x and

a remainder ( )r x . We must write this as ( ) ( )

( )( ) ( )

f x r xq x

d x d x .

Method 1)

Using Long Division: Divide 4 22 5y y y by 2 1y y

Divide 3 2 2 8x x x by 1x

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Let 3 2( ) 3 2 2 5f x x x x

1. Use long division to divide ( )f x by 2x

What is the Quotient? ________ What is the Remainder? ________

2. Use Synthetic Substitution to evaluate (2)f . _________

How is (2)f related to the remainder? _____________________.

What do you notice about the other constants in the last row of the synthetic

substitution? _________________________________________

The remainder theorem says that when a polynomial 𝑓(𝑥) is divided by a linear (1st degree)

polynomial, 𝑥 − 𝑐 and you solve it for x so it is in the form 𝑥 = 𝑐, if you evaluate 𝑓(𝑐), the

answer will be your remainder.

Example:

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The factor theorem is an extension of the remainder theorem. Recall that the remainder

theorem states that if you evaluate a polynomial with the c-value found by setting the linear

𝑥 + 𝑐 = 0, the result will be the remainder of the polynomial had we done either long or

synthetic division. The factor theorem extends that by saying if the remainder theorem

results in 0, the linear 𝑥 + 𝑐 must be a factor of the polynomial.

Example:

𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑤ℎ𝑒𝑡ℎ𝑒𝑟:

𝑎) 𝑥 + 1 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑓(𝑥) = 𝑥4 − 5𝑥2 + 6𝑥 − 1

𝑏) 𝑥 − 2 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑓(𝑥) = 𝑥3 − 3𝑥2 + 4

The remainder is ____. This means that 𝑥 − 2 is a factor

of 𝑓(𝑥) = 𝑥3 − 3𝑥2 + 4Therefore you can write the

result as:

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Example: How many zeros does the polynomial have? 𝑓(𝑥) = (𝑥 − 3)(𝑥 − 1)2(𝑥 + 2)3

Factoring Polynomials

Factor: 3 2( ) 3 13 2 8f x x x x given that ( 4) 0f

Solving Polynomials (which also means Finding the ________)

One zero of 3 2( ) 6 3 10f x x x x is 5x . Find the other zeros of the function.

𝑓(𝑥) = 𝑥5 − 4𝑥4 − 7𝑥3 + 14𝑥2 − 44𝑥 + 120 if 2, 5, and -3 are factors. Find the other zeros of the function.

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Using Polynomial Division in Real Life

A company that manufactures CD-ROM drives would like to increase its production. The demand function for

the drives is 275 3p x , where p is the price the company charges per unit when the company produces x

million units. It costs the company $25 to produce each drive.

a) Write an equation giving the company’s profit as a function of the number of CD-ROM drives it

manufactures.

b) The company currently manufactures 2 million CD-ROM drives and makes a profit of $76,000,000. At

what other level of production would the company also make $76,000,000?

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2.6 Finding Rational Zeros

We call the list of all 𝑝

𝑞 “possible” or “potential” rational zeros.

Find the rational zeros of 3 2( ) 4 11 30f x x x x .

List the possible rational zeros:

Test (Verify zero using the Remainder Theorem)

Factor

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Find the all real zeros of: 4 3 2( ) 15 68 7 24 4f x x x x x .

Solving Polynomial Equations in Real Life

A rectangular column of cement is to have a volume of 20.253.ft The base is to be square, with sides 3 ft. less

than half the height of the column. What should the dimensions of the column be?

A company that makes salsa wants to change the size of the cylindrical salsa cans. The radius of the new can

will be 5 cm. less than the height. The container will hold 144∏ 3cm of salsa. What are the dimensions of the

new container?

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2.7 Finding All Zeros of Polynomial Function

Use Zeros to write a polynomial function

Example 1: Find all the zeros of 5 4 2( ) 2 8 13 6f x x x x x

Example 2: Write a polynomial function f(x) of least degree that has real coefficients, a

leading coefficient of 1, and 2 and 1 + 𝑖 as zeros.

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Example 3: Write a polynomial function f(x) of least degree that has real coefficients, a

leading coefficient of 2, and 5 and 3𝑖 are zeros.

2.8 Analyzing Graphs of Polynomial Functions -Using the Graphing Calculator

1) Approximate Zeros of a Polynomial Function. [2nd][TRACE][zero] or if you make 𝑦2 = 0 then you can

find the intersection of the two equations. [2nd][TRACE][intersect]

2) Find Maximum and Minimum Points of a Polynomial Function. [2nd][TRACE][maximum or minimum]

3) Find a Polynomial Model that fits a given set of data. (Cubic, Quartic Regression) and make predictions.

[STAT, EDIT, input data in 𝐿1 and 𝐿2, STAT, CALC, CubicReg or QuartReg, VARS, Y-VARS, Function 𝑌1]

The graph of a function has ups and down or peaks and valleys. A peak is known as a maximum (plural -

maxima) and a valley is termed as a minimum (plural - minima) of given function. There may be more

than one maximum or minimum in a function. The maxima and minima are collectively known as

extrema (whose singular is extremum) that are said to be the largest and smallest values undertaken

by given function at some point either in certain neighborhood (relative or local extrema) or over the

domain of the function (absolute or global extrema).

How many turning points does a polynomial have?

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Never more than the degree minus 1, we say that a polynomial has at most 1n turning points.

Recall that the degree of a Polynomial with one variable is the largest exponent of that variable.

Example: a polynomial of Degree 4 will have 3 turning points or less

x4−2x2+x

has 3 turning points

x4−2x

has only 1 turning point

The most is 3, but there can be less.

We can look at a graph and count the number of turning points and simply add 1 and that will be the least

degree of the polynomial.

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Identify the zeros (x-intercepts), maximums and minimums of 3 2( ) 2 5 1f x x x x and

4 3 2( ) 2 5 4 6f x x x x

A rectangular piece of sheet metal is 10 in. long and 10 in. wide. Squares of side length x are cut from

the corners and the remaining piece is folded to make an open top box.

a) What size square can be cut from the corners to give a box with a volume of 25 cubic inches.

b) What size square should be cut to maximize the volume of the box? What is the largest possible

volume of the box?

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An open box is to be made from a rectangular piece of cardboard that is 12 by 6 feet by cutting out squares of

side length x ft from each corner and folding up the sides.

a) Express the volume of the box ( )v x as a function of the size x cut out at each corner.

b) Use your calculator to determine what size square can be cut from the corners to give a box with a

volume of 40 cubic feet.

c) Use your calculator to approximate the value of x which will maximize the volume of the box.

Use your Graphing Calculator to find the appropriate polynomial model that fits the data. Use it to make

predictions.

x 1 2 3 4 5 6 ….. 10

f(x) 26 -4 -2 2 2 16 ?

The table shows the average price (in thousands of dollars) of a house in the Northeastern United States for

1987 to 1995. Find a polynomial model for the data. Then predict the average price of a house in the

Northeast in 2000.

x 1987 1988 1989 1990 1991 1992 1993 1994 1995

f(x) 140 149 159.6 159 155.9 169 162.9 169 180

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Polynomial Inequalities with degree two or more. 1. Set one side of the inequality equal to zero. 2. Temporarily convert the inequality to an equation.

3. Solve the equation for x . If the equation is a rational inequality, also determine the values of x where the expression is undefined (where the denominator equals zero). These are the partition values.

4. Plot these points on a number line, dividing the number line into intervals. 5. Choose a convenient test point in each interval. Only one test point per interval is needed. 6. Evaluate the polynomial at these test points and note whether they are positive or negative.

7. If the inequality in step 1 reads 0 , select the intervals where the test points are positive. If the inequality

in step 1 reads 0 , select the intervals where the test points are negative.

Quadratic Inequalities

Example 1. Solve. a) 2 2 15 0x x b) 2 2 15 0x x c) 23 11 1 5x x More Polynomial Inequalities

Example 2. Solve.

a) ( 5)(3 4)( 2) 0x x x b) 316 0x x c) 2( 5) ( 1) 0x x

d) 3 23 16 48x x x e) 2 2(2 3) 0x x