Polynomials, Linear Factors, and Zeros To analyze the actored orm o a polynomial. To write a polynomial unction rom its zeros. Describe the relationship among solutions, zeros, - intercept, and actors. Factor theorem, multiple zero, multiplicity, relative maimum, relative minimum
Now we know that P() is a polynomial unction, the solutions o the related polynomial equation P ()= are the zeros o the unction. Finding the zeros o the polynomial will help you actor the polynomial, graph the unction, and solve the related polynomial equation Problem : Writing a Polynomial in Factored Form What is the actored orm o GCF: Factor 5 Your turn What is the actored orm o Answer: 4 5 5 5
Take a note Roots, Zeros, and -intercepts The ollowing are equivalent statements about a real n n number b and a polynomial P an an... a a b is a linear actor o the polynomial P(). b is a zero o the polynomial unction y=p(). b is a root(or a solution) o the polynomial equation P()= b is an -intercept o the graph o y=p()
Problem : Finding zeros o a polynomial unction What are the zeros o? Graph the unction. y Step : Use the zero-product property to ind zeros.,,,,, Step : Find points or -values between the zeros. Evaluate 4 6 8 4,,6,8
Step : Determine the end behavior. The unction is a cubic unction. The coeicient o is. So the end behavior is down and up. Step 4: Use the zeros, the additional points and end behavior to sketch the graph.,,,,,,,8,6, 4 end behavior is down and up.
Your turn What are the zeros o y 5?Graph the unction Answer: (-,) (-5,) (,) (,)
Factor Theorem The epression (-a) is a actor o a polynomial i and only i the value a is a zero o the related polynomial unction A theorem is a statement that has been proven on the basis o previously established statements, such as other theorems, and previously accepted statements.
Problem : Writing a polynomial unction rom its zeros A. What is the cubic polynomial unction in standard orm with zeros -,, and? Write a linear actor or each zero 5 6 5 6 5 6 Multiply (-) and (-) Distributive Property 4 Simpliy The cubic polynomial has zeros -,, and. B. What is a quartic unction in standard orm with zeros -,-, and g 4 5 6 4 4 4 4 4 4 4 The quartic polynomial has zeros -,-,, and. 4 4 4
C. Graph both unctions. How do the graphs dier? How are they similar? g Both have -int. at -, and. The cubic has down and up end behavior. The quartic has up and up behavior. The cubic unction has two turning points and it cross the -ais at -. The quartic unction touches the -ais at - but doesn t cross it. The quartic unction has three turning points. Note: you can write the polynomial unction in the actored orm as and g in the g()) the repeated linear actor + makes - a multiple zero. Since the linear actor + appear twice, you can say that - is a zero multiplicity. in general, a is a zero o multiplicity n means +a appears n times as a actor.
Your turn a)what is a quadratic polynomial unction with zeros and -? Answer: 9 b) What is a cubic polynomial unction with zeros,, and -? Answer: - 9 7 c) Graph both unctions. How do the graphs dier? How are they similar? Both graphs have -int. o and -.The quadratic has up and up and one turning point, the cubic has down and up end behavior and two turning points
Note: i a is a zero o multiplicity n in the polynomial unction y then the behavior o the graph at the -intercept a will be close to linear i n=, close to quadratic i n=, close to cubic i n=. and so on. I a is a zero o even multiplicity then the graph o its unction touches the -ais at a and turns around. I a is a zero o odd multiplicity then the graph o its unction crosses the -ais at a Problem 4: Finding multiplicity o a zero. P() 4 What are the zeros o 8? What are their multiplicities? How does the graph behave at these zeros? 4 8 8 4 Factor out the GCF then actor 8
Since the number is a zero o multiplicity. The numbers - and 4 are zeros o multiplicity The graph looks close to linear at the -intercepts - and 4. It resembles a parabola at -intercept I the graph o a polynomial unction has several turning points, the unction can have a relative maimum and relative minimum. A relative maimum is the value o the unction at an up to down turning point. A relative minimum is the value o the unction at a down to up turning point. 4 8
Your turn What are the zeros o 4 4? What are their multiplicities? How the graph behave at this zeros? Answer: 4 4 or is a zero o multiplicity,the graphs looks close to linear at =; is a zero o multiplicity, the graph looks close To quadratic at =
Your turn again What are the relative maimum and minimum o 4 Answer: Using a graphing calculator the maimum is 8 at =-4 And the relative minimum is -8 at = The relative maimum is the greatest y-value in the neighborhood o the -value. The maimum at a verte o a parabola is the greatest y-value o all the -values
Problem 5: Using a polynomial Function to Maimize Volume The design o a digital camera maimizes the volume while keeping the sum o the dimensions at 6 inches. I the length must be.5 times the height, what each should each dimension be? Step : Deine a variable Step : Determine length and width l. 5 l w h 6.5 w.5 w 6 w 6.5 6 Step : Model the volume V V lwh.56.5.75 9 Let be =the height o the camera Step 4: Graph it. I using Graphing calculator use the MAXIMUM eature to ind that the maimum value is 7.68 or a height.6 h.6, l.5.5(.6).4, w 6.5 6.5(.6)
Your turn A designer wants to make a rectangular prism bo with maimum volume, while keeping the sum o its length, width, and height equal to in. The length must be times the height. What should each dimension be? Answer: Step : Deine a variable Let h Step : Determine length and width length= width V w Step : Model the volume l w 4 w w h 4 ( )( 4) 6 Step 4: Graph it. Height= Length= 6 Width= 4
Class work odd Homework even TB pgs 9-94 eercises 7-44