50 Unit 4: Polynomial and Rational Functions Polynomial Functions A polynomial function y px ( ) is a function of the form p( x) ax + a x + a x +... + ax + ax+ a n n 1 n n n 1 n 1 0 where an, an 1,..., a, a1, a0are real constants and are called the coefficients of px ( ). n is the degree of px ( ) and is a positive integer. an is called the leading coefficient and a0 is the constant term of the polynomial. The domain of any polynomial is all real numbers. Ex. 1 Determine the degree, the leading coefficient and the constant term of the polynomial. a) 4 5x + 7x x+ 7 b) gx ( ) 1x + 5x 4x End Behavior of a Polynomial There are four scenarios: 1) Sketch px ( ) x, px ( ) x ( n is even, a n > 0 ) 4 ) Sketch px ( ) x, px ( ) x ( n is even, a n < 0 ) 4 As x, px ( ) As x, px ( ) As x, px ( ) As x, px ( )
51 ) Sketch px ( ) x, px ( ) x ( n is odd, a n > 0 ) 5 4) Sketch px ( ) x, px ( ) x ( n is odd, a n < 0 ) 5 As x, px ( ) x, px ( ) As x, px ( ) x, px ( ) As x and x, the graph of the polynomial p( x) ax + a x + a x +... + ax + ax+ a resembles the graph of n n 1 n n n 1 n 1 0 y n ax n. Ex. Use the zeros and the end behavior of the polynomial to sketch an approximation of the graph of the function. a) x 9x b) gx x x 4 ( ) 5 + 4
5 c) + 5 x x Repeated Zeros If a polynomial has a factor of the form ( x c) k, where k > 1, then x cis a repeated zero of multiplicity k. If k is even, the graph of flattens and just touches the x -axis at x c. If k is odd, the graph of flattens and crosses the x -axis at x c. Ex. 4: Sketch the given graphs x x + x 4 gx x x x ( ) ( 1) ( + )( )
5 Ex. 5: The cubic polynomial px ( ) has a zero of multiplicity two at x 1, a zero of multiplicity one at x, and p( 1). Determine px ( ) and sketch the graph. Ex. 6: An open box is to be made from a rectangular piece of cardboard that is 1 by 6 feet by cutting out squares of side length x feet from each corner and folding up the sides. a) Express the volume of the box vx ( ) as a function of the size x cut out at each corner. b) Use your calculator to approximate the value of x which will maximize the volume of the box. Ex. 7: The difference of two non-negative numbers is 10. What is the maximum of the product of the square of the first number and the other?
54 The Intermediate Value Theorem Suppose that f is continuous on the closed interval [ aband, ] let N be any number between f( a) and f( b ), where f( a) f( b). Then there exists a number c in ( ab, ) such that f() c N. Ex. 1: Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. x + x 1, [0,5], f( c ) 11 Ex. : Show that there is a root of the equation x + x 1 0in the interval (0,1).
55 The Division Algorithm Let and d( x) 0be polynomials where the degree of is greater than or equal to the degree of d( x ). Then there exists unique polynomials qx ( ) and rx ( ) such that rx ( ) qx ( ) + or d( xqx ) ( ) + rx ( ). d( x) d( x) where rx ( ) has a degree less than the degree of d( x ). Ex. 1: Divide the given polynomials. a) 6x 19x + 16x 4 x b) x 1 x 1 c) x x x+ 6 x + 1
56 Remainder Theorem If a polynomial is divided by a linear polynomial x c, then the remainder r is the value of at x c. In other words, f() c r Ex. : Use the Remainder Theorem to find r when 4x x + 4is divided by x. Ex. : Use the Remainder Theorem to find f() c for 4 x 5x + 7 when 1 c Synthetic Division Synthetic division is a shorthand method of dividing a polynomial px ( ) by a linear polynomial x c. It uses only the coefficients of px ( ) and must include all 0 coefficients of px ( ) as well. Ex. 4: Use synthetic division to find the quotient and remainder when a) f x ( ) x 1 is divided by x 1 b) 4 x 14x + 5x 9 is divided by x + 4 c) 4 8x 0x x 8x + + is divided by 1 x 4
57 Ex. 5: Use synthetic division and the Remainder Theorem to find f() c for 6 5 4 x + 4x + x 8x 6x + 9 when c. Ex. 6: Use synthetic division and the Remainder Theorem to find f() c for x 7x + 1x 15 when c 5. The Factor Theorem A number c is a zero of a polynomial px ( ) ( pc () 0) if and only if ( x c) is a factor of px ( ). Examples: Determine whether a) x + 1is a factor of f x x x + x 4 ( ) 5 6 1 b) x is a factor of x x + 4
58 Fundamental Theorem of Algebra A polynomial function px ( ) of degree n > 0 has at least one zero. In fact, every polynomial function px ( ) of degree n > 0 has at exactly n zeros. Complete Factorization Theorem Let c1, c,... c n be the n (not necessary distinct) zeros of the polynomial function of degree n > 0 : p( x) ax + a x + a x +... + ax + ax+ a. n n 1 n n n 1 n 1 0 Then px ( ) can be written as the product of n linear factors p( x) a ( x c )( x c ) ( x c ). n 1 n Ex. 1: Give the complete factorization of the given polynomial px ( ) with given information: a) px ( ) x 9x + 6x 1; 1 x is a zero. b) 4 px ( ) 4x 8x 61x + x+ 15 ; x, x 5 are both zeros.
59 c) px ( ) x 6x 16x+ 48 ; ( x ) is a factor. d) + ; x(x 1) is a factor. 4 px ( ) x 7x 5x x Ex. : Find a polynomial function of degree three, with zeros 1,-4, 5 such that the graph possesses the y - intercept (0,5).
60 The Rational Zero Test Suppose p q is a rational zero of 1 ( ) n n f x ax a x a x n + + +... + ax + ax+ a, n n 1 n 1 0 where a0, a1..., an are integers and an 0. Then p divides a0 and q divides a n. The Rational Zero Test provides a list of possible rational zeros. Examples: Find all the rational zeros of then factor the polynomial completely. a) f x x x x x 4 ( ) 10 + 8. b) f x x x x x 4 ( ) + +
61 Complex Roots of Polynomials Consider factoring the function: f x ( ) x 1 The Square Root of -1 We define i 1 so that i 1. Complex Numbers A complex number is a number of the form a + bi where a and b are real numbers. The number a is called the real part and the number b is called the imaginary part. Complex Arithmetic Ex. 1: a) ( + i) (6 i) b) ( + i)(4 i) c) ( 6 i)( + 6 i) d) (4 5 i)(4 + 5 i) Complex Conjugates The complex conjugate for a complex number z a + bi is z a bi. In general, ( a bi)( a + bi)
6 Ex. : Simplify. a) ( + i) (1 6 i) ( i) (1+ 7 i) Ex. : Simplify. a) 4 b) 8 Ex. 4: Determine all solutions to the equation x 4x+ 1 0 Ex. 5: Completely factor f x ( ) x 1.
Ex. 6: Find the complete factorization of multiplicity two. 6 4 x 1x + 47x 6x+ 6 given that 1 is a zero of Conjugate Pairs of Zeros of Real Polynomials If the complex number z a + bi is a zero of some polynomial px ( ) with real coefficients, then its conjugate z a bi is also a zero of px ( ). Ex. 7: Find a rd degree polynomial gx ( ) with real coefficients and a leading coefficient of 1 with zeros 1 and 1 i. 4 Ex. 8: 1+ i is a zero of x x 4x + 18x 45. Find all other zeros and then give the complete factorization of f( x ).
64 Rational Functions A rational function y is a function of the form functions. px ( ), where p and q are polynomial qx ( ) Ex. 1: Recall the parent function 1. Use transformations to sketch x gx ( ) x 1 Asymptotes of Rational Functions The line x ais a vertical asymptote of the graph of if or as x a + (from the right) or x a (from the left). Vertical Asymptotes px ( ) The graph of has vertical asymptotes at the zeros of qx ( ) after all of the common factors qx ( ) of px ( ) and qx ( ) have been canceled out; the values of x where qx ( ) 0and px ( ) 0. Holes The graph of px ( ) has a hole at the values of x where qx ( ) 0and px ( ) 0. qx ( )
65 Horizontal Asymptotes The line y bis a horizontal asymptote of the graph of if bwhen x or x. In particular, with a rational function n px ( ) ax n + a x +... + ax+ a m q( x) b x + b x +... + bx+ b m n 1 n 1 1 0 m 1 m 1 1 0 There are three cases: 1. If n< m, then y 0 is the horizontal asymptote. x Ex: 1x + 7x an. If n m, then y is the horizontal asymptote. b m Ex: x + 6x 4x + x. If n> m, then there is no horizontal asymptote. Ex: 4 x x 5x + 1 x 4x Slant Asymptote If the degree of numerator is exactly one more than the degree of the denominator, the graph of has a slant asymptote of the form y mx + b. The slant asymptote is the linear quotient found by dividing px ( ) by qx ( ) and essentially disregarding the remainder. Example: x 8x+ 1 x + 1
66 Ex. : Find all asymptotes and intercepts and sketch the graphs of the given rational functions: a) x 1 Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept: b) x + x + 4 Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept:
67 x + c) ( x )( x+ 5) Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept: d) x x x + 1 Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept: x + 1 e) x x Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept:
68 f) (x+ 1)( x ) ( x )( x+ 1) Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept: Ex. : Sketch the graph of a rational function that satisfies all of the following conditions: + as x 1 and as x 1 as x and as x has a horizontal asymptote y 0 has no x -intercepts Has a local maximum at ( 1, ) + Ex. 4: The product of two non-negative numbers is 60. What is the minimum sum of the two numbers?
Honors Precalculus Academic Magnet High School 69 Name Mandelbrot Set Activity using Fractint fractal generator STEP 1 - CREATE, SAVE, and PRINT an inspirational, visually pleasing area of the Mandelbrot set. Important Menu Items: VIEW- Image Settings, Zoom In/ Out box, Coordinate Box FRACTALS- Fractal Formula, Basic Options, Fractal Parameters COLORS- Load Color- Map FILE- Save As 1) Start Fractint by clicking on the desktop icon. Fractint always starts with the Mandelbrot set, but in case things get weird, ALWAYS make sure mandel is selected in the Fractal-Fractal Formula menu item. Use the Image Settings box to set the size of the picture (800 x 600 should work fine). ) Use the Zoom In/Out feature along with the Colors-Load Color Map to create a variation of the Mandelbrot set. If the color palettes do not load, double click on the box that is labeled Pallette Files (*.Map) If you zoom in a few times you lose detail, you can increase the iterations in the Fractals-Basic Options Box- Remember that the more iterations the computer has to perform, the longer it will take ) Use the Fractals-Fractal Params window to record the x and y mins and maxs of the viewing rectangle on the imaginary plane. 4) Using the Coordinates box, point your arrow to a point you think is in the Mandelbrot set and record the x and y values. 5) Repeat #4 for a point you think is NOT in the set. 6) SAVE the fractal. Write down the coordinates (x and y mins and maxs) and number of iterations of your current position in the Mandelbrot set. 7) Print your fractal. STEP - Create a typed text document (1 page or so) including, but not limited to: The NAME of your group s fractal and the name of everyone in your group A short story about your creation (what it makes you think of, color, choice, etc.)
70 STEP Typed: 1) List the x and y mins and maxs for your viewing rectangle from Step 1 ) Recall the coordinates of the point you thought was in the Mandelbrot set from Step 1. Let x a and y b for the complex number a + bi Let this number a + bi c iterate this value 100 or more times using the Mandelbrot sequence: x 0 c x 1 x 0 + c x x 1 + c Etc You will be using decimals and your calculator. Unlike the fractals, these calculations will not be pretty. Let your TI-84 do the work for you (i is above the decimal point). ) Record the last 0 iterations for analysis. Remember that you may need to scroll the TI- 84 to the right to get the entire number 4) Were your predictions right about this point? Do you need more information to determine if it is in the set? 5) Repeat for the point you thought was not in the set. 6) Summarize your findings. TURN IN ALL STEPS PAPER-CLIPPED together in order. Extra Credit: Create your own color map. http://www.nahee.com/spanky/www/fractint/fractint.html - for info on Fractint