Polynomials, Linear Factors, and Zeros What you’ll learn To analyze the factored form of a polynomial. To write a polynomial function from its zeros. Describe the relationship among solutions, zeros, x- intercept, and factors. Vocabulary Factor theorem, multiple zero, multiplicity, relative maximum, relative minimum

Now we know that P(x) is a polynomial function, the solutions of the related polynomial equation P (x)=0 are the zeros of the function. Finding the zeros of the polynomial will help you factor the polynomial, graph the function, and solve the related polynomial equation. Problem 1: Writing a Polynomial in Factored Form What is the factored form of GCF: x Factor Your turn What is the factored form of Answer:

Roots, Zeros, and x-intercepts Take a note Roots, Zeros, and x-intercepts The following are equivalent statements about a real number b and a polynomial is a linear factor of the polynomial P(x). b is a zero of the polynomial function y=P(x). b is a root(or a solution) of the polynomial equation P(x)=0 b is an x-intercept of the graph of y=P(x)

Problem 2: Finding zeros of a polynomial function What are the zeros of ? Graph the function. Step 1: Use the zero-product property to find zeros. Step 2: Find points for x-values between the zeros. Evaluate

Step 3: Determine the end behavior. The function is a cubic function. The coefficient of is 1. So the end behavior is down and up. Step 4: Use the zeros, the additional points and end behavior to sketch the graph. end behavior is down and up.

Your turn What are the zeros of ?Graph the function Answer: (-3,32) (-5,0) (0,0) (3,0)

The expression (x-a) is a factor of a polynomial if and only Factor Theorem The expression (x-a) is a factor of a polynomial if and only if the value a is a zero of the related polynomial function A theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements.

Problem 3: Writing a polynomial function from its zeros A. What is the cubic polynomial function in standard form with zeros -2,2, and 3? Write a linear factor for each zero Multiply (x-2) and (x-3) Distributive Property Simplify The cubic polynomial has zeros -2,2, and 3. B. What is a quartic function in standard form with zeros -2,-2,2 and 3 The quartic polynomial has zeros -2,-2,2, and 3.

C. Graph both functions. How do the graphs differ? How are they similar? Both have x-int. at -2,2 and 3. The cubic has down and up end behavior. The quartic has up and up behavior. The cubic function has two turning points and it cross the x-axis at -2. The quartic function touches the x-axis at -2 but doesn’t cross it. The quartic function has three turning points. Note: you can write the polynomial function in the factored form as and in the g(x)) the repeated linear factor x+2 makes -2 a multiple zero. Since the linear factor x+2 appear twice, you can say that -2 is a zero multiplicity 2. in general, a is a zero of multiplicity n means x+a appears n times as a factor.

Your turn a)What is a quadratic polynomial function with zeros 3 and -3? Answer: b) What is a cubic polynomial function with zeros 3,3, and -3? c) Graph both functions. How do the graphs differ? How are they similar? Both graphs have x-int. of 3 and -3.The quadratic has up and up and one turning point, the cubic has down and up end behavior and two turning points

Problem 4: Finding multiplicity of a zero. Note: if a is a zero of multiplicity n in the polynomial function then the behavior of the graph at the x-intercept a will be close to linear if n=1, close to quadratic if n=2, close to cubic if n=3. and so on. If a is a zero of even multiplicity then the graph of its function touches the x-axis at a and turns around. If a is a zero of odd multiplicity then the graph of its function crosses the x-axis at a Problem 4: Finding multiplicity of a zero. What are the zeros of ? What are their multiplicities? How does the graph behave at these zeros? Factor out the GCF then factor

Since the number 0 is a zero of multiplicity 2. The numbers -2 and 4 are zeros of multiplicity 1 The graph looks close to linear at the x-intercepts -2 and 4. It resembles a parabola at x-intercept 0 If the graph of a polynomial function has several turning points, the function can have a relative maximum and relative minimum. A relative maximum is the value of the function at an up to down turning point. A relative minimum is the value of the function at a down to up turning point. Relative minimum

Your turn What are the zeros of ? What are their multiplicities? How the graph behave at this zeros? Answer: 0 is a zero of multiplicity 1,the graphs looks close to linear at x=0; 2 is a zero of multiplicity 2, the graph looks close To quadratic at x=2

Your turn again What are the relative maximum and minimum of Answer: Using a graphing calculator the maximum is 80 at x=-4 And the relative minimum is -28 at x=2 The relative maximum is the greatest y-value in the neighborhood of the x-value. The maximum at a vertex of a parabola is the greatest y-value of all the x-values

Problem 5: Using a polynomial Function to Maximize Volume The design of a digital camera maximizes the volume while keeping the sum of the dimensions at 6 inches. If the length must be 1.5 times the height, what each should each dimension be? Step 1: Define a variable Let be x=the height of the camera Step 2: Determine length and width Step 3: Model the volume Step 4: Graph it. If using Graphing calculator use the MAXIMUM feature to find that the maximum value is 7.68 for a height 1.6

A designer wants to make a rectangular prism box with Your turn A designer wants to make a rectangular prism box with maximum volume, while keeping the sum of its length, width, and height equal to 12 in. The length must be 3 times the height. What should each dimension be? Step 1: Define a variable Let Step 2: Determine length and width length= width Step 3: Model the volume Answer: Step 4: Graph it. Height=2 Length= 6 Width= 4

Class work odd Homework even TB pgs 293-294 exercises 7-44