Introduction to Polynomials(Sections 8-1 & 8-3)PEARSON TEXT PP.340-345 AND PP.350-356.PORTIONS OF THE POWERPOINT TAKEN FROM: HOLT ALGEBRA 2: INVESTIGATING GRAPHS OF POLYNOMIAL FUNCTIONS

Essential QuestionHow do you determine the end behaviors of a polynomial function?

VOCABULARYDEFINE THE TERMS ON TEXT P.340 (L8-1) AND TEXT P.350 (L8-3). ADD THESE INTO YOUR VOCABULARY LIST IN YOUR BUFF BINDER.

• Degree of a Monomial• Degree of a Polynomial• End Behavior• Monomial• Polynomial• Polynomial Function• Standard Form of a Polynomial Function• Turning Point

• Multiple Zero• Multiplicity• Relative Minimum• Relative Maximum• Root

KEY understanding from yesterday’s discussion – A polynomial has ONLY whole number exponents!!!(no negative, no decimal, no fractional exponents)

The LEAD Coefficient is the coefficient of the highest degreed term… “4” for the example above. The LEAD coefficient indicates the RIGHT-Side behavior of the graph (what the graph will be doing as the function approaches positive infinity!!).

Classifying PolynomialsDegree Name using Degree Polynomial Example Number

of TermsName using Number of Terms

0 Constant 1 Monomial

1 Linear 2 Binomial

2 Quadratic 1 Monomial

3 Cubic 3 Trinomial

4 Quartic 2 Binomial

5 Quintic 4 Polynomial of 4 terms

Sooo…You need to know the definition for each of the following terms…

Constant function Linear function Quadratic function Cubic function Quartic function Quintic function

Monomial Binomial trinomial

Constant term Linear term Quadratic term Cubic term Quartic term Quintic term

To classify a polynomial, it must first be SIMPLIFIED and put into STANDARD FORM! That means performing any needed multiplication and combining ALL like terms.

Title your paper Intro to Polynomials and answer the following questions. Part I Practice:

(This is 2-7 on page 344 in the textbook)

Simplify & put each into Standard Form. Then…

End behavior is a description of the values of the function as x approaches infinity (x +∞) or negative infinity (x –∞). The degree and leading coefficient of a polynomial function determine its end behavior. It is helpful when you are graphing a polynomial function to know about the end behavior of the function... What the function will be doing at the extreme “ends” of the graph.

0 turns 4 turns3 turns2 turns1 turn

What is the relationship between the END BEHAVIORS of an odd-degreed polynomial?

What is the relationship between the END BEHAVIORS of an even-degreed polynomial?

OPPOSITE DIRECTIONS

SAME DIRECTION

The LEAD coefficient, therefore, determines the RIGHT-SIDE end behavior…

What is the RIGHT-SIDE behavior of a graph that has a positive lead coefficient?

UP (y increases as x approaches positive infinity)

What is the RIGHT-SIDE behavior of a graph that has a negative lead coefficient?

DOWN (y decreases as x approaches positive infinity)

Example 1A: Determining End Behavior of Polynomial Functions

Identify the leading coefficient, degree, and end behavior.

A. Q(x) = –x4 + 6x3 – x + 9 The leading coefficient is –1, which is negative. The degree is 4, which is even. As x –∞, P(x) –∞, and as x +∞, P(x) –∞.

LEFT-side end behavior is when x approaches negative infinity (x –∞)

RIGHT-side end behavior is when x approaches positive infinity (x +∞)

Example 1B: Determining End Behavior of Polynomial Functions

Identify the leading coefficient, degree, and end behavior.

B. P(x) = 2x5 + 6x4 – x + 4 The leading coefficient is 2, which is positive. The degree is 5, which is odd. As x –∞, P(x) –∞, and as x +∞, P(x) +∞.

Left side is going down… Right side is going up…

Example 2A: Determining End Behavior of Polynomial Functions

Identify the leading coefficient, degree, and end behavior.

a. P(x) = 2x5 + 3x2 – 4x – 1 The leading coefficient is 2, which is positive. The degree is 5, which is odd. As x –∞, P(x) –∞, and as x +∞, P(x) +∞.

You Try…

Left side is going down… Right side is going up…

Example 2B: Determining End Behavior of Polynomial Functions

Identify the leading coefficient, degree, and end behavior.

b. S(x) = –3x2 + x + 1 The leading coefficient is –3, which is negative. The degree is 2, which is even. As x –∞, P(x) –∞, and as x +∞, P(x) –∞.

You Try…

Left side is going down… Right side is going down…

Example 3A: Using Graphs to Analyze Polynomial FunctionsIdentify whether the function graphed has an odd or even degree and a positive or negative leading coefficient.

As x –∞, P(x) +∞, and as x +∞, P(x) –∞.P(x) is of odd degree with a negative leading coefficient.

Right side is going down so Lead Coefficient is negative… Left side is doing the opposite so it has to be an Odd-degreed function…

Example 3A: Using Graphs to Analyze Polynomial FunctionsIdentify whether the function graphed has an odd or even degree and a positive or negative leading coefficient.

As x –∞, P(x) +∞, and as x +∞, P(x) +∞.P(x) is of even degree with a positive leading coefficient.

Example 4A: Using Graphs to Analyze Polynomial FunctionsIdentify whether the function graphed has an odd or even degree and a positive or negative leading coefficient.

You Try…As x –∞, P(x) +∞, and as x +∞, P(x) –∞.P(x) is of odd degree with a negative leading coefficient.

Example 4B: Using Graphs to Analyze Polynomial FunctionsIdentify whether the function graphed has an odd or even degree and a positive or negative leading coefficient.

You Try…As x –∞, P(x) +∞, and as x +∞, P(x) +∞.P(x) is of even degree with a positive leading coefficient.

A turning point is where a graph changes from increasing to decreasing or from decreasing to increasing. A turning point corresponds to a “local” or “relative” maximum or minimum.

In other words… a relative maximum is the highest point on a “hump” of a graph (where the graph is turning down as we go to the right).

In other words… a relative minimum is a lowest point on a “hump” of a graph (where the graph is turning up as we go to the right).

A polynomial function of degree n has at most n – 1 turning points and at most n x-intercepts. If the function has n distinct roots, then it has exactly n – 1 turning points and exactly n x-intercepts. You can use a graphing calculator to graph and estimate any relative maximum and minimum values.

Example:Q(x) = –x4 + 6x3 – x + 9

Degree: 4Most number of turning points: 3Most number of x-intercepts: 4

Example 5A: Determine Maxima and Minima with a CalculatorGraph f(x) = 2x3 – 18x + 1 on a calculator, and estimate the local maxima and minima.

Step 1 Graph.The graph appears to have one local maxima and one local minima.

Step 2 Find the maximum.Press to access the CALC menu. Choose 4:maximum.The local maximum is approximately 21.7846.

–5

–25

25

5Choose a “LEFT bound” and a “RIGHT bound” that surround the point… Then hit ENTER when it says “Guess”

Example 5A: Determine Maxima and Minima with a Calculator (cont)Graph f(x) = 2x3 – 18x + 1 on a calculator, and estimate the local maxima and minima.

Step 3 Find the minimum.Press to access the CALC menu. Choose 3:minimum. The local minimum is approximately –19.7846.

Choose a “LEFT bound” and a “RIGHT bound” that surround the point… Then hit ENTER when it says “Guess”

Example 5B: Determine Maxima and Minima with a CalculatorGraph g(x) = x3 – 2x – 3 on a calculator, and estimate the local maxima and minima.

Step 1 Graph.The graph appears to have one local maxima and one local minima.

Step 2 Find the maximum.Press to access the CALC menu. Choose 4:maximum.The local maximum is approximately –1.9113.–5

–5

5

5

Example 5B: Determine Maxima and Minima with a Calculator (cont)Graph g(x) = x3 – 2x – 3 on a calculator, and estimate the local maxima and minima.

Step 3 Find the minimum.Press to access the CALC menu. Choose 3:minimum.The local minimum is approximately –4.0887.

Example 6: Determine Maxima and Minima with a CalculatorGraph h(x) = x4 + 4x2 – 6 on a calculator, and estimate the local maxima and minima.

You Try… Step 1 Graph.The graph appears to have one local maxima and one local minima.Step 2 There appears to be no maximum.

–10

–10

10

10

Step 3 Find the minimum.Press to access the CALC menu. Choose 3:minimum.The local minimum is –6.

Determine the degree, the end behavior, and the local minima and maxima of each polynomial function. Draw a rough sketch of each function and label the parts.

Part II: Continue working on your

Intro to Polynomials

paper.

(This is 17-25 on pages 344-345 in the textbook)

Root Form of a Polynomial Function

P(x) is said to be in its’ ROOT FORM if it is completely factored!

P(x) = a (x – r1)(x – r2)(x – r3)…(x – rn)

where r1, r2, r3…rn are all roots of the polynomial.

EXAMPLE: f(x) = x (x – 2)(x – 3)(x + 5) is the root form of a 4th degree (Quartic) polynomial.

It has roots of 0, 2, 3 & –5 …since solving each of the factors produces these solutions! …Also, these x-values will be the x-intercepts of the graph!

Step 2 Find the turning points (any relative minimums and/or maximums… It has a MAX at about (–0.8, 8.2) and a MIN at about (2.1, –4.1).

You Try…Write a cubic polynomial function that has zeros of 1, 2 & –3 …

Translate the ZEROS into the FACTORS… (x – r)… and write out the Root Form. f(x) = (x – 1)(x – 2)(x + 3)

Now multiply the 1st two polynomials… (x – 1)(x – 2) = x2 – 2x – 1x + 2) = x2 – 3x + 2 f(x) = (x2 – 3x + 2)(x + 3)

Now multiply the result by the 3rd polynomial… x3 – 3x2 + 2x + 3x2 – 9x + 6… f(x) = x3 – 7x + 6

Now you have the polynomial function in Standard Form!

The “multiplicity” of a zero is “how many times that linear factor is repeated in a polynomial’s factored form.”

P(x) = x3(x – 1)(x – 1)(x + 4)(x – 7) = x3(x – 1)2(x + 4)(x – 7)

… so “0” is a zero of multiplicity 3… “1” is a zero with a multiplicity of 2… : ”–4” & “7” are zeros, each of multiplicity 1.

This a 7th degree polynomial since all the multiplicities of the zeros add to 7. ( 3 + 2 + 1 + 1 = 7)

Part III: Continue working on your Intro to Polynomials paper.

(Do the following problems from pages 354-355 in the textbook)

Find the zeros of each function. Identify their multiplicity. Then sketch a graph of the function with all significant points.

Find the zeros of each function. Identify their multiplicity.

Write a polynomial function in standard form with the given zeros.

29. x = –5, –5, 1 30. x = 3, 3, 3

33. x = 0, 0, 2, 3