Warm UP Solve the following equations: (x – 7)(x + 4) = 0 x3 + 8 = 0

Warm UP Solve the following equations: (x – 7)(x + 4) = 0 x3 + 8 = 0

Unit 3: Polynomial Functions
Test October 5th

3-1 Solving polynomials Evaluate a polynomial using the Remainder Theorem. Use the Factor Theorem to solve a polynomial equation. Use the Rational Root Theorem to completely factor and solve a polynomial

Fundamental theorem of algebra
Any polynomial of degree n has n roots - but some roots may be complex numbers. A root (or zero) is where the polynomial is equal to zero. So, a polynomial of degree 3 will have 3 roots. A polynomial of degree 4 will have 4 roots. And so on.

Example: What are the roots of x2 − 9? x2 − 9 has a degree of 2 so there are 2 roots. Solve it to find the roots by setting it equal to zero: x2 − 9 = 0 x = ±3 So the roots are −3 and +3 The roots are c1 = -3 and c2 = +3 so the factors are: x2 − 9 = (x+3)(x−3) The Linear Factors are (x+3) and (x−3)

How many roots will the polynomial x3−1 have? Factor this to find the roots! SOAP x3−1 = (x−1)(x2 + x + 1) The roots are x = 1 and there are 2 imaginary roots as well. How can we find those? How many times do you think this polynomial will touch the x-axis?

Fundamental Theorem Of Algebra
A Polynomial Equation of the form P(x) = 0 of degree ‘n’ with complex coefficients has exactly ‘n’ roots. These roots can be: all real all imaginary or a combination of both real and imaginary

Real or Imaginary Roots?
So, if a polynomial has ‘n’ roots will its graph have ‘n’ x-intercepts? In this example, the degree n = 3, and if we factor the polynomial, the roots are x = -2, 0, 2. We can also see from the graph that there are 3 x-intercepts.

Real or Imaginary Roots?
NO! Just because a polynomial has ‘n’ roots doesn’t mean that they are all REAL (which means they will touch the x-axis)! In this example, the degree is still n = 3, but there is only one Real x-intercept or root at x = -1, the other 2 roots must have imaginary components.

So, how do we find those roots?
Factoring Quadratic formula Complete the square Taking roots Graphing

Factoring Polynomials
Expressions are Factors of a Polynomial if, when they are multiplied, they equal that original polynomial: (x - 3) and (x + 5) are factors of the polynomial x = 3 and x = -5 are roots of the polynomial

Factors vs. Roots Factors: What multiplies together to get the original polynomial. They are usually grouped in parenthesis. ex., (x+2), (2x-3), (x-8) Roots: What the x values equal after setting the factor equal to 0. Can also be called zeros, solutions, or x-intercepts. ex., x=-2, x=3/2, x=8

Since Factors are a Product...
…and the only way a product can equal zero is if one or more of the factors are zero… …then the only way the polynomial can equal zero is if one or more of the factors are zero. This is called the Zero Product Property.

Factor Theorem If remainder = 0 when we plug a # (a root) into some function, then (x – #) is a factor. We can get a remainder using long division or synthetic division, or the remainder theorem. We want a remainder of 0 to find the other roots…and we want to be able to find other factors and roots!

Solving Polynomials graphically
Suppose we want to find the roots of Graph this on your calculator. By inspecting the graph of the function, find one of the solutions. Since we know one solution to f(x), we also know a factor of the expression. This means that if we divide f(x) by the factor there will be a remainder of zero.

Remainders can be useful!
The remainder theorem states: If the polynomial f(x) is divided by (x – c), then the remainder is f(c). This leads us to the factor theorem: For a polynomial, f(c) = 0 if and only if x – c is a factor of f(x). Or in other words, If f(c) = 0, then x – c is a factor of f(x). If x – c is a factor of f(x), then f(c) = 0. If we know a factor, we know a zero! If we know a zero, we know a factor!

Example Use the Remainder Theorem to find the indicated function value. Is x – 3 a factor of f(x)?

Example Use the Remainder Theorem to find the indicated function value. Is x = 2 a zero of f(x)?

ASSIGNMENT DO THE FRONT ONLY

Warm UP

Find Roots/Zeros of a Polynomial
Sometimes, polynomials are easier to solve than others. It will take practice until you will be able to find the *best* way to solve them. In this case, we can find the zeros of this polynomial by setting it equal to 0 and factoring. The roots are: 0, -2, 2

Finding Roots/Zeros of Polynomials
We will use the Fundamental Theorem Of Algebra and Descartes’ Rule of Signs to predict the nature of the roots of a polynomial in a similar manner as we use the discriminant to predict the roots of a quadratic polynomial. We use skills such as factoring, polynomial division and the quadratic formula to find the zeros/roots of polynomials.

Given the polynomial below, determine the total number of roots it will have: If we cannot factor the polynomial, but know one of the roots, we can use division to break down the polynomial. (x - 5) is a factor We can solve the resulting polynomial to get the other 2 roots:

Remainder theorem Completely factor f(x) = x4 - 6x3 - 4x + 24 if you know that f(6) = 0.

Chapters 1 & 2 The Conservative Tradition in Educational Thought
Factor theorem If (x-1) is a factor, find all other factors of g(x) = 3x3 + 8x2 – 3x – 8

Factor theorem Given that (x-2) and (x+3) are factors of
find the remaining factors.

Given that (x-10) is a factor of factors. Then determine the roots.
You Try!! Given that (x-10) is a factor of , find the other factors. Then determine the roots.

Now, complete the BACK of the practice from YESTERDAY 

Warm UP 9/17/18 1) Find the value of k that makes the linear expression (x-1) a factor of the cubic expression 𝑥 3 +3 𝑥 2 −5𝑥+𝑘. 2) If the remainder that the polynomial 𝑥 3 +6 𝑥 2 +𝑘𝑥 −1 leaves upon division by (x-1) is 766, what is the value of k?

Solving Polynomials Remember the factor and remainder theorems we learned last week. Finish the front and the back of the worksheet. I will stop you at 9:15 (and I might collect this for a grade!) We have one additional thing to learn today!

What if you are just asked to solve a polynomial without knowing anything else? We know this polynomial will have 3 roots. What else can we do?

Descartes’ Rule of Signs
Arrange the terms of the polynomial P(x) in standard form The number of times the coefficients of the terms of P(x) change sign tells you the number of Positive Real Roots (or less by any even number) The number of times the coefficients of the terms of P(-x) change sign tells you the number of Negative Real Roots (or less by any even number) In the examples that follow, we will use Descartes’ Rule of Signs to predict the number of + and - Real Roots!

Find the Roots of a Polynomial
For higher degree polynomials, finding the roots (real and imaginary) is easier if we know one of the roots. Descartes’ Rule of Signs can help get you started. Complete the table below:

HOMEWORK!

Warm UP Determine the number of possible positive and negative real solutions and imaginary solutions for the polynomial MAKE SURE IT’S IN STANDARD FORM! Please turn in your homework 

Solving polynomials Solving polynomials can be tricky business sometimes. So far, we have solved when we have a “starting point” – usually one of the factors. Sometimes, a graphing utility can be a helpful tool to identify some roots, but in general there is no simple formula for solving other polynomials like the quadratic formula aids us in solving quadratics. There is however a tool that we can use for helping us to identify Rational Roots of the polynomial in question.

The Rational Root Theorem
The Rational Root Theorem is another tool to predict the values of Rational Roots:

List the Possible Rational Roots
For the polynomial: All possible values of: All possible Rational Roots of the form p/q:

Narrow the List of Possible Roots
For the polynomial: Descartes’ Rule: All possible Rational Roots of the form p/q:

Find a Root That Works For the polynomial:
Substitute each of our possible rational roots into f(x). If a value, a, is a root, then f(a) = 0. (Roots are solutions to an equation set equal to zero!)

Find the Other Roots Now that we know one root is x = 3, do the other two roots have to be imaginary? What other category have we left out? To find the other roots, divide the factor that we know into the original polynomial:

Find the Other Roots (con’t)
The resulting polynomial is a quadratic, but it is not factorable. Solve the quadratic by either setting it equal to zero and using the quadratic formula (any time), or by isolating the x and taking the square root of both sides (missing b term).

Rational root theorem

Find all the roots: (Hint: Rational root theorem)

Find all the roots:

Descartes’ Rule of Signs

Find all zeros.

Investigation Look at the following functions on your graphing calculator and use the graph to fill in the table: Function Zeros Cross Bounce Degree

Multiplicity This refers to the number of times the root is a zero of the function. We can have “repeated” zeros. Odd Multiplicity: f crosses the x-axis at its root; f(x) changes signs Even Multiplicity: f “kisses” or is tangent to the x-axis at it root; f(x) doesn’t change signs

Multiplicity All four graphs have the same zeros, at x = 6 and x = 7, but the multiplicity of the zero determines whether the graph crosses the x-axis at that zero or if it instead turns back the way it came.

Polynomial Functions Multiplicity
The number of times a factor (m) of a function is repeated is referred to its multiplicity (zero multiplicity of m). Multiplicity of an Even Exponent The graph of the function touches the x-axis but does not cross it. Think of this as a “bounce” Multiplicity of an Odd Exponent The graph of the function crosses the x-axis.

Multiplicity Identify the zeros and their multiplicity 1. 3. 1. 2. 1.
3 is a zero with a multiplicity of 1. Graph crosses the x-axis. -2 is a zero with a multiplicity of 3. Graph crosses the x-axis. -4 is a zero with a multiplicity of 1. Graph crosses the x-axis. 7 is a zero with a multiplicity of 2. Graph touches the x-axis. -1 is a zero with a multiplicity of 1. Graph crosses the x-axis. 4 is a zero with a multiplicity of 1. Graph crosses the x-axis. 2 is a zero with a multiplicity of 2. Graph touches the x-axis.

How many zeros are there in the polynomial h(x) = (x + 2)7(x2 – 25)5, counting multiplicity?

multiplicity Sometimes a factor appears more than once. That is its Multiplicity. Example: x2−6x+9 x2−6x+9 = (x−3)(x−3) "(x−3)" appears twice, so the root "3" has Multiplicity of 2 The Multiplicities are included when we say "a polynomial of degree n has n roots".

Find all zeros.

Warm UP (then you have a quiz!)
Given f(x)= x4 –22x3 +39x2 +14x+120 , answer the following: How many zeros are there for this polynomial in the set of complex numbers? Is x – 20 a factor of x4 –22x3 +39x2 +14x+120? Find all other roots of the polynomial.

Objective: To use graphs to make statements about functions.
LG 3-2 Analyzing Graphs of Polynomials Objective: To use graphs to make statements about functions.

Vocabulary End Behavior: the value of f(x) as x approaches positive and negative infinity Multiplicity: the number of times a root occurs at a given point of a polynomial equation.

Write the factored form of a polynomial function that crosses the x-axis at x = –2 and x = 5 and touches the x-axis at x = 3. Which of the zeros of the function must have a multiplicity greater than 1? Explain your reasoning. Write two additional polynomial functions that meet the same conditions as described above.

Graphs of polynomials are smooth and continuous.
No gaps or holes, can be drawn without lifting pencil from paper No sharp corners or cusps This IS the graph of a polynomial This IS NOT the graph of a polynomial

More NON-examples of graphs of non-polynomial functions

Graphs of Polynomials The graphs of polynomials of degree 0 or 1 are lines. The graphs of polynomials of degree 2 are parabolas. The greater the degree of the polynomial, the more complicated its graph can be. The graph of a polynomial function is always a smooth curve; that is it has no breaks or corners.

Let’s look at the graph of where n is an even integer.
and grows steeper on either side Notice each graph looks similar to x2 but is wider and flatter near the origin between –1 and 1 The higher the power, the flatter and steeper

Let’s look at the graph of where n is an odd integer.
Notice each graph looks similar to x3 but is wider and flatter near the origin between –1 and 1 and grows steeper on either side The higher the power, the flatter and steeper

Investigation Look at the graphs for each function on your GDC and observe the end behavior for the polynomial functions. What does the degree of the polynomial function tell you about its end behavior? Function Left End Right End Degree

and RIGHT LEFT END BEHAVIOR OF A GRAPH The degree of the polynomial along with the sign of the coefficient of the term with the highest power will tell us about the left and right hand behavior of a graph.

Even degree polynomials rise on both the left and right hand sides of the graph (like x2) if the coefficient is positive. Any additional terms may cause the graph to have some turns near the center but will always have the same left and right hand behavior determined by the highest powered term. left hand behavior: rises right hand behavior: rises

Even degree polynomials fall on both the left and right hand sides of the graph (like - x2) if the coefficient is negative. turning points in the middle left hand behavior: falls right hand behavior: falls

Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x3) if the coefficient is positive. turning Points in the middle right hand behavior: rises left hand behavior: falls

Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like x3) if the coefficient is negative. turning points in the middle left hand behavior: rises right hand behavior: falls

End Behavior

End Behavior the behavior of the graph as x gets very large (approaches positive infinity) OR as x gets very small (or approaches negative infinity). Notation: The very far left end of a graph: The very far right end of a graph: Which term of the polynomial function is most important when determining the end behavior of the function?

End Behavior Degree Leading Coefficient End Behavior Even Positive
Negative Odd

Indicate if the degree of the polynomial function shown in the graph is odd or even and indicate the sign of the leading coefficient

Polynomial Functions Write an equation for a possible function:
What is the degree of this function? Describe the end behavior:

What a possible equation for the graph:

Homework: Page 357 #1-9

Warm Up! 4/26/18

Turning points of polynomial functions
What is the difference between local and absolute maxima and minima? Polynomial functions have turning points corresponding to local maximum and minimum values. The y – coordinate of a turning point is a local or relative maximum if the point is higher than all nearby points. The y – coordinate of a turning point is a local or relative minimum if the point is lower than all nearby points. Global or absolute minimums and maximums are the greatest or least values of the entire function.

Polynomial Functions Turning Points
The point where a function changes directions from increasing to decreasing or from decreasing to increasing. If a function has a degree of n, then it has at most n – 1 turning points. If the graph of a polynomial function has t number of turning points, then the function has at least a degree of t + 1 . What is the most number of turning points the following polynomial functions could have? 3-1 2 5-1 4 8-1 7 12-1 11

So, the lowest degree of this
More about those turning points… This graph has 3 turning points. The lowest degree of a polynomial is (# turning points + 1). So, the lowest degree of this polynomial is 4 ! Relative maximum The graph “turns” Relative minimums The graph “turns”

What’s happening? 4 5 Relative Maximums Relative Minimums
As x  - , f(x) 4 The number of turning points is _____ . As x  + , f(x) 5 The lowest degree of this polynomial is _____ . The leading coefficient is __________ . positive

determine the following:
Sign of the leading coefficient: The real roots and their multiplicity: The number of turning points: The lowest possible degree: The relative minimum The relative maximum The absolute maximum The absolute minimum

You Try Sign of the leading coefficient:
The real roots and their multiplicity: The number of turning points: The lowest possible degree: The relative minimum The relative maximum The absolute maximum The absolute minimum

Vocabulary: Domain: The set of values of the independent variable(s) for which a function or relation is defined. Typically, this is the set of x-values that give rise to real y-values. Range: The resulting y-values we get after substituting all the possible x-values. Intervals of increase and decrease: The portions of the domain where the function is getting larger or smaller, respectively.

Other Characteristics:
Domain Range Intervals of increase and decrease Symmetry

Example 1 Determine the following characteristics: The zeros
The extrema The domain The range the intervals of increase The intervals of decrease

Symmetry Notes

WE WILL TALK ABOUT THE CARDS AFTER THIS
Warm UP 4/27/18 Sketch a graph of the polynomial and determine the following characteristics: The zeros (&their multiplicity) The # of extrema The absolute extrema The domain The range The y-intercept

Card sort In your groups, sort the cards.
Write the NUMBER of the card on the paper ONLY. Please DO NOT write ON the cards 

Steps for Graphing a Polynomial
Determine end behavior; add arrows to the graph to help guide your sketch Determine maximum number of turning points in graph by subtracting 1 from the degree. Use this info to guide your sketch. Find and plot y intercept by putting 0 in for x. Find the zeros by setting polynomial = 0 and solving. Plot these on the graph Determine multiplicity of zeros. Use this info to guide your sketch. Join the points together in a smooth curve.

Test REVIEW: Study these!
LG 5-1: Polynomial Operations LG 5-2: Solving Polynomials LG 5-3: Analyzing Polynomials Division Long Synthetic Vocabulary Factoring Binomial expansion Operations Remainder Theorem Rational Root Theorem Factor Theorem Descartes Theorem Fundamental Theorem Multiplicity Turning Points End behavior Characteristics Sketching Graphs

Let’s graph:

f(x) =x(x-3)2(x2+4)

Warm up 4/30/18 Domain: Range: Increasing Interval(s):
Decreasing Interval(s): x-intercept(s): y-intercept(s): Relative Maximum(s): Relative Minimum(s): Absolute Maximum(s): Absolute Minimum(s): Minimum Degree: Sign of Leading Coefficient: Exact equation of the function:

1) 2) 3)

Binomial expansion The coefficient of x3y5 in the expansion of (ax + y)8 is 3,584. What is the value of a?

How do we find the factors and roots of Polynomials?
To find the Roots/Zeros of Polynomials, you will use: The Fundamental Theorem of Algebra Descartes’ Rule of Signs The Rational Root Theorem

Find all zeros.

Irrational Root Theorem
For any polynomial if is a root, then is also a root Irrationals always come in pairs. Real values do not. These are called CONJUGATES

Examples: 1. If a polynomial has a root Other roots
Degree of Polynomial 2 2. If a polynomial has roots x = -1, x = 0, Other roots Degree of Polynomial 6

Imaginary Root Theorem
Complex Numbers The complex number system includes real and imaginary numbers. Standard form of a complex number is: a + bi. a and b are real numbers. Conjugate Pairs Theorem

Examples 1) A polynomial function of degree three has 2 and 3 + i as it zeros. What is the other zero? 2) A polynomial function of degree 5 has 4, 2 + 3i, and 5i as it zeros. What are the other zeros? 3) A polynomial function of degree 4 has 2 with a zero multiplicity of 2 and 2 – i as it zeros. What are the zeros?

Find the remaining complex zeros of the given polynomial functions

Find the zeros of the given polynomial functions

Vieta’s Formulas Although many polynomials have very complicated roots, we can often determine a lot about them using Vieta's Formulas (named after François Viète). Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

Polynomials Newport Math Club Warm up Problems:
7 coins are placed on a table. Initially there are 4 heads and 3 tails facing upwards. You desire to separate this group of 7 coins into 2 groups (not of the same size) with each group having an equal number of tails. Obviously, this would only be possible if you’re allowed to flip some of the coins, and you are. What is your strategy in accomplishing this? Oh yeah, you’re blindfolded. Generalize. What is your strategy when there are x number of heads and y number of tails facing upwards? title and warm up

Translating Polynomial Functions
Consider the cubic function How can we graph The graph is transformed with a horizontal translation of 2 to the left and a vertical translation of 3 down.

Representing Polynomials Task
ESSENTIAL QUESTIONS: What is the relationship between graphs and algebraic representations of polynomials? What is the connection between the zeros of polynomials, when suitable factorizations are available, and graphs of the functions defined by polynomials? What is the connection between transformations of the graphs and transformations of the functions obtained by replacing f(x) by f(x + k), f(x) + k, -f(x), f(-x)?

Given the polynomial function of the form:
If k is a zero, Zero: __________ Solution: _________ Factor: _________ If k is a real number, then k is also a(n) __________________. x = k x = k (x – k) x - intercept

What kind of curve? All polynomials have graphs that are smooth continuous curves. A smooth curve is a curve that does not have sharp corners. Sharp corner – This graph must not be a polynomial function. A continuous curve is a curve that does not have a break or hole. Hole Break This is not a continuous curve!

End Behavior (Think of a line with positive slope!)
LC < 0 , Odd Degree (Think of a line with negative slope!) LC > 0 , Even Degree (Think of a parabola graph… y = x2 .) LC < 0 , Even Degree (Think of a parabola graph… y = -x2 .) LC > 0 , Odd Degree y x y x y x y x As x  - , f(x) As x  - , f(x) As x  - , f(x) As x  - , f(x) As x  + , f(x) As x  + , f(x) As x  + , f(x) As x  + , f(x)

Graphing by hand 3 2 Step 1: Plot the x- and y-intercepts
Step 2: End Behavior? Number of Turning Points? Negative-odd polynomial of degree 3 Example #1: Graph the function: f(x) = -(x + 4)(x + 2)(x - 3) and identify the following. End Behavior: _________________________ Degree of polynomial: ______________ # Turning Points: _______________________ As x  - , f(x) As x  + , f(x) 3 2 You can check on your calculator!! x-intercepts

Graphing with a calculator
Example #2: Graph the function: f(x) = x4 – 4x3 – x2 + 12x – 2 and identify the following. Positive-even polynomial of degree 4 End Behavior: _________________________ Degree of polynomial: ______________ # Turning Points: _______________________ y-intercept: _______ As x  - , f(x) As x  + , f(x) 4 3 Relative max (0, -2) Plug equation into y= Find minimums and maximums using your calculator Real Zeros Relative minimum Absolute minimum

Graphing without a calculator
Example #3: Graph the function: f(x) = x3 + 3x2 – 4x and identify the following. Positive-odd polynomial of degree 3 End Behavior: _________________________ Degree of polynomial: ______________ # Turning Points: _______________________ As x  - , f(x) As x  + , f(x) 3 2 1. Factor and solve equation to find x-intercepts f(x)=x(x2 + 3x – 4) = x(x - 4)(x + 1) 2. Plot the zeros. Sketch the end behaviors.

Polynomial Functions: Real Zeros, Graphs, and Factors (x – c)
If P is a polynomial function and c is a real root, then each of the following is equivalent. (x – c) __________________________________ . x = c __________________________________ . (c, 0) __________________________________ . is a factor of P is a real solution of P(x) = 0 is a real zero of P is an x-intercept of the graph of y = P(x)

TEST DAY 5/1/18

Warm UP Solve the following equations: (x – 7)(x + 4) = 0 x3 + 8 = 0

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Warm UP Solve the following equations: (x – 7)(x + 4) = 0 x3 + 8 = 0

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