Concept Category 4. Polynomial Functions
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1 Concept Category 4 Polynomial Functions
2 (CC1) A Piecewise Equation 2 ( x 4) x 2 f ( x) ( x 3) 2 x 1
3 The graph for the piecewise
4 Polynomial Graph (preview)
5 Still the same transformations
6 CC4 Learning Targets Polynomial operations: Add & Subtract ; multiplication (distribution); division Polynomial graph: Roots (x-intercepts); y-intercept; local maximum and minimum points Leading degree and leading coefficient and Endbehavior How to change a standard form of a polynomial equation to a factored form, then sketch the factored form
7 Adding Polynomials Some people prefer to add polynomials by stacking them. If you choose to do this, be sure to line up the like terms! (x 2 + 3x + 1) + (4x 2 +5) (x 2 + 3x + 1) + (4x 2 +5) 5x 2 + 3x + 6 Stack and add these polynomials: (2a 2 +3ab+4b 2 ) + (7a2+ab+-2b 2 ) (2a 2 + 3ab + 4b 2 ) (2a 2 +3ab+4b 2 ) + (7a2+ab+-2b 2 ) + (7a 2 + ab + -2b 2 ) 9a 2 + 4ab + 2b 2
8 Adding Polynomials Add the following polynomials; you may stack them if you prefer: 1) 3x 3 7x 3x 3 4x 6x 3 3x 2) 2w 2 w 5 4w 2 7w 1 6w 2 8w 4 a 3 4a 3 3) 2a 3 3a 2 5a 3a 3 3a 2 9a 3
9 Subtracting Polynomials Subtract: (3x 2 + 2x + 7) - (x 2 + x + 4) Step 1: Change subtraction to addition (Keep-Change-Change.). (3x 2 + 2x + 7) + (- x x + - 4) Step 2: Underline OR line up the like terms and add. (3x 2 + 2x + 7) + (- x x + - 4) 2x 2 + x + 3
10 Subtracting Polynomials Subtract the following polynomials by changing to addition (Keep-Change-Change.), then add: 1) x 2 x 4 3x 2 4x 1 2x 2 3x 5 2y 2 y 9 2) 9y 2 3y 1 7y 2 4y 10 g 3 3g 2 3 3) 2g 2 g 9 g 3 g 2 g 12
11 x x 2 x x 2x On your notebook x x x x x x x x x x x x x x x x x x x x x x x x x x x x cannot ( different exponent s) x x x 3x x x x
12 Polynomial Multiplication A. (2x 3)( x 4) B. 3(2x 1)( x 3) C. 2 x(x 3)(3x 2) D. (x 3) * it is not x 9 E. ( x 1)( x 2)( x 3) F. ( x 2) * it is not x 8 3
13 Solutions 2 A. 2x 11x 12 2 B. 3(2x 5x 3) 2 2 6x 15x 9 C. 2 x(3x 7x 6) D x 14x 12x ( x 3)( x 3) x 2 6x 9
14 E. ( x 1)( x 2)( x 3) 2 ( x 3x 2)( x 3) x 3x 3x 9x 2x x 6x 11x 6 F. ( x 2) 3 ( x 2)( x 2)( x 2) 2 ( x 4x 4)( x 2) x 2x 4x 8x 4x x 6x 12x 8
15 Dividing Polynomials
16 Powers Decrease 5 2 4x 2 2xxxxx 1xx x 20x xxx x y 6xxxxyyyyyyyy 1 1 6x y 6xxxxxyyyyyyyyyy xyy xy
17 dividing a polynomial by a monomial r s 3rs 9r s 1. 3rs 6r 2 s 2 3rs 3rs2 3rs 9r 2 s 3rs 2rs s 3r
18 Simplify a b 6a b 18ab 3ab 3a2 b 3ab 6a3 b 2 3ab 18ab 3ab a 2a 2 b 6
19 Simplify x y 3x 3x 12x2 y 3x 3x 3x 4xy 1
20 Long Division Remember your elementary school math, long division? or
21 Let s solve these together Your childhood problem: Your teenage-hood problem: x x x x If the remainder is not 0, the binomial factor outside is not a real factor for the equation, move onto the next factor.
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23 Long Division - divide a polynomial by a polynomial Think back to long division from 3rd grade. How many times does the divisor go into the dividend? Put that number on top. Multiply that number by the divisor and put the result under the dividend. Subtract and bring down the next number in the dividend. Repeat until you have used all the numbers in the dividend.
24 ex) x 2 5x 24 x 3 x 2-8 x 3 x 5x 24 -( x 2 + 3x ) x 2 /x = x - 8x x/x = -8 -(- 8x - 24 ) 0
25 ex) 3x 2 4x 10 x 2 3x 2-2 x 2 3x 4x 10 -( 3x 2 +6x ) 3x 2 /x = 3x -2x/x = -2-2x +10 -(- 2x -4 ) 14 R.
26 ex) h h h h 2 + 4h 3 2 h 4 h 0h 11h 28 -( h 3-4h 2 ) h 4 Negative power also means division h 3 /h = h 2 4h 2 /h = 4h 4h 2-11h -( 4h 2-16h ) 5h ( 5h - 20 ) 5h/h = 5 48
27 Happy Wednesday! CC3 CC4 For each equation: find the Perform the operations roots, the vertex, y- intercept, and sketch 2 2 1] y 2( x 5) 7 2 2] y 2x x 3 1] (12 x 2x 3) 2x 2] Divide x 3 1 by x - 1 Challenge : Divide then factor the quotient 3 2 3](4x 12x 19x 12) (2x 1)
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30 Divide x 3 1 by x - 1 x 1 x 2 x 3 + x 0x x x 3 - x 2 x 2 + 0x - 1 x 2 - x - + x - 1 x - 1
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33 x y x
34 Polynomials: 3 Forms 2/15/18 Linear : a( x) 3x 5 b( x) 2( x 3) Quadratic : f x x x 2 ( ) g( x) 2( x 3)(2x 5) h x 2 ( ) 4( x 3) 5 Cubic : w x x x 3 ( ) A( x) 4 x( x 3)( x 4) 3 B(x) 5(x 4) 6
35 Polynomial: Standard Form Ex 1) y 3x 5x 2x 4x x 7 Leading Coefficient: this effects the End-Behavior The highest power. Leading Degree. Constant: y-intercept value (0, -7) The leading degree tells you how many factors, therefore how roots you are supposed to have: in this case you are supposed to have 5 factors, thus 5 roots
36 Polynomial: Factored Form Ex 2) (x+0) y 3 x( x 2)( x 3)( x 1)( x 6) Leading Coefficient: this effects the End-Behavior How many factors indicate the leading degree: so 5 factors here means degree 5; also 5 roots Constant is not obvious for this form
37 Convert Standard Form to Factored Form ex) 3 2 f ( x) 2x 8x 2x 12 a) What s the leading coefficient? b) What s the leading degree? How many factors do we have? How many roots do we have? c) How do we convert it to factored form if one of the factors is x+3?
38 Convert Standard Form to Factored Form ex) 3 2 f ( x) 2x 8x 2x 12 a) What s the leading coefficient? 2 b) What s the leading degree? 3 How many factors do we have? 3 How many roots do we have? 3 a) How do we convert it to factored form if one of the factors is x+3? y = 2(x+3)(x+2)(x-1)
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40 Try it Again Turn your work in before you leave the class 3 2 g( x) 2x 2x 28x 48 a) What s the leading coefficient? b) What s the leading degree? How many factors do we have? How many roots do we have? c) How do we convert it to factored form if one of the factors is x-4?
41 Answer: y = -2(x+2)(x+3)(x-4)
42 Happy Tuesday 2/20 th Quick Check this Friday 3 2 1) y 2x 3x 2x 3 given factor : 2x 3 Find the factored form, then find all the roots 2 2) y 2x 8x 2 Find the y-intercept, the roots, and the factored form
43 More on Roots (x-intercepts) If you plug any of the roots back into the equation (x s), you should get 0 for y
44 When that first factor is not given : Rational Roots Example: 3 2 w( x) 2x x 22x 21 Step 1: Find all possible combination by using the factors of the constant divided by the factors of the leading coefficient 1, 21, 3, 7 1, 2 Step 2: simplify , 3, 7, 21,,,, Step 3: if any of these numbers is an actual root (x-intercept) for the equation, you should get y= 0 when you plug it into the equation (x s)
45 3 2 y 2x x 22x 21 x x x x 3 2 1, 2(1) (1) 22(1) , 2( 1) ( 1) 22( 1) , 2(3) (3) 22(3) , 2( 3) ( 3) 22( 3) 21 We can continue to perform this operation until we find all the roots. but we already have enough information to figure out the rest
46 Step 4: convert the root to factor, then perform long division, then perform factoring or Quadratic Formula x x 1 ( x 1) 3 ( x 3) Which factor do you want to use for long division? Your choice (or you can use both) x x x x x x x x
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52 Now you try 3 2 example 2) y 3x 8x 7x 12 Find the factored form and find all the roots Whether the leading coefficient or the constant is positive or negative does not matter: you have to include both positive and negative when finding its rational roots
53 Solution 3 2 example 2) y 3x 8x 7x 12 y ( x 3)( x 1)(3x 4) or y 3(x 3)(x 1)(x 1.33) x 3, 1,1.33
54 Quick Check result tomorrow or Wednesday Who was absent on Friday? QC make up today
55 Graphing Polynomial Monday 2/26th
56 How many equations can you find here? What are they? (think transformation)
57 Remember piecewise function from CC1? Pay attention to the vertex point of each graph (horizontal shift)
58 How about this one?
59 3 2 y 0.007( x 6) ( x 2) ( x 2) Leading Coefficient
60 To find the Leading Coefficent a 1. Pick a point on the graph line 2. Never use the x-intercept points (or any point too close to the x-intercepts) 3. Plug-into the equation to solve a
61 How about this one?
62 CC1 piecewise function f( x) 2 ( x 4) x 2 or (, 2] ( x 3) 2 x 1 or [1, )
63 So how about this one?
64 Still the same parent graphs + transformation
65 2 2 y a( x 4) ( x 3)
66 A polynomial equation is similar to a piece-wise equation, But no intervals (x limitations) 2 2 y a( x 4) ( x 3) The Leading Coefficient
67 Goal Problem: Key Features Domain : Range : End Behavior : Intervals of Increase : Decrease : x intercepts : y intercept : Factored Form : Standard Form :
68 D : (, ) R : [ 3.3, ) EB : Left x, y Right x, y Increase at 2 x 1, 1.3 x Decrease at x 2, 1 x 1.3 x int ( 2,0) (0,0) (2,0) y int (0,0)
69 To find the factored form of the polynomial graph 1) What are the parents? 2) What is the vertex (x-intercept) of each? Think Piecewise!
70 y a( x 2) 2 ( x)( x 2)
71 Factored Form of the Equation: The exponents are also called multiplicities y a( x 2) 2 ( x)( x 2) Leading Coefficient You have to solve for it Factors Find a point that is not an x-intercept, usually we can use the y-intercept point if it is not also an x-intercept
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73 Don t Forget X-intercepts are also called
74 Domain : Range : End Behavior : Intervals of Increase : Decrease : x intercepts : y intercept : Factored Form : Standard Form : Key Features
75 Key Features Domain : (, ) Range :(, ) End Behavior : Lx, y ; Rx, y Intervals of Increase :(,0) (1.4, ) Decrease : (0,1.4) x intercepts :( 0.6, 0) (1,0) (1.6,0) y intercept : (1,0) Factored Form : y a(x 0.6)(x 1)( x 1.6) solve a Standard Form :
76 Find the key features
77 Domain : Range : End Behavior : Intervals of Increase : Decrease : x intercepts : y intercept : Factored Form : Standard Form : Key Features
78 Domain : (, ) Range : (,7.2] End Behavior : Left x y Right x y Intervals of Increase : (, 5) U ( 3,1) Decrease :( 5, 3) U (1,5) U (5, ) x intercepts :( 6,0) ( 3,0) (5,0) y intercept : ( 0,6.8) Factored Form : StandardForm : 1 y a(x 6) (x 3) 2 (x 5) 3
79 2 3 y a( x 6)( x 3) ( x 5) ( 5,4)
80 Find the Key Features ( 3.5, 4) Think Piecewise! (horizontal)
81 Domain : Range : End Behavior : Intervals of Increase : Decrease : x intercepts : y intercept : Factored Form : Standard Form : Key Features
82 There are 4 equations here
83 y a ( x 3)( x 1)( x)( x 2) To solve a : use ( 3.5, 4)
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