Name: Period: Pre-Cal AB: Unit 1: Polynomials Monday Tuesday Block Friday 11/1 1 Unit 1 TEST Function Operations and Finding Inverses 16 17 18/19 0 NO SCHOOL Polynomial Division Roots, Factors, Zeros and Graphs of Polynomials Modeling Data Fundamental Theorem March Binomial Theorem 4 Mixed Practice Binomial Theorem 5/6 Unit 1 TEST Part 1 (60 pts) 4/5 Unit 1 TEST Part (40 pts) Start new unit 7 will get new packet Binomial Theorem Lesson #1: Function Operations I can Determine the degree, leading coefficient, and constant of a polynomial. Determine the inverse of a function and/or verify inverses of functions. Determine the sum, difference, product, and quotient of functions. Determine domains of functions and composite functions. I. Polynomials A. Poly means so a polynomial has terms. B. The degree is determined by the highest power. Linear is a power of. Quadratic is a power of. Cubic is a power of. Quartic is a power of. Quintic is a power of. After that we say th power. C. The leading coefficient and constant is determined in Standard Form. 4 5 D. Example f ( x) = x + x 8x 0x II. Operations A. To add/subtract functions you must. B. To multiply functions, straight. C. To divide functions, the second function and. D. The domain for +/-/x will be the same but division may be different. E. To determine the domain you may recall: III. Composite Functions In a fraction,. In a square root,. Otherwise it is all real numbers. (, ) A. Notation: They may give you f(g(x)) or ( f g )( x) B. If functions are inverses then f(g(x)) = x and g(f(x)) = x. IV. Inverse Function A. An inverse undoes the function. It switches (x, y) to (y, x) B. Interchange the x and the y. (make y x and make x y) C. Resolve for y. D. You may need to do the SAT trick x = becomes y = y x 1 E. Write in function notation f ( x). They mean the same thing.
V. Model Problems Guided Practice Given f ( x) = x 1; g( x) = x Find (f + g)(x) Find (f g)(x) On Your Own f g Find ( x) Find (fg)(x) Given f ( x) = x + and (f g)(x), 1 g( x) = x, find (f + g)(x)and f Find (fg)(x) and ( x) g Find the inverse: f ( x) = x Find the inverse: f x ( ) = x + 6 Find the inverse: 4 f ( x) = x 5 Find the inverse: f ( x) = 6 x Verify 5 4x + 5 f ( x) =, g( x) = x 4 x are inverses. Verify f(x) = x 1 and x g( x) = are inverses.
Restrict the domain so that f(x) is a function. ( x ) f ( x) = Restrict the domain so that f(x) is a function. f ( x) = x + Practice #1 f g 1 4: Given the functions, find (f + g)(x), (f g)(x), (fg)(x), and ( x) f ( x) = x + f ( x) = x + x 1) ) ) g( x) = 4x + 7 g( x) = x 1 f ( x) = 9x + 5 9) Given the functions, evaluate each: g( x) = x h( x) = x 6 then state the domain for each. f ( x) = x + 4) x 1 g( x) = x + f ( x) = x + g( x) = 5) f(g(4)) 6) ( h g )( x) 7) ( f g )(1) 8) f(f(0)) 9) ( h f )( x) 1 x + 10 1) Find the inverse of each function. x 10) f(x) = ½ x 11) f ( x) = 1) x f x ( ) 5x 4 = 1) f ( x) = 5 + x Verify that each are inverses. 14) f ( x) = x ; g( x) = x 1 15) f x x g x x ( ) = 1; ( ) = Lesson #: Long and Synthetic Division I can Divide with a linear divisor. Divide with a non-linear divisor. Apply Remainder Theorem to determine the remainder using synthetic substitution. Apply Factor Theorem to determine if (x a) is a factor of a polynomial. I. Synthetic Division may be used only when dividing by a term in the form of x c. A. Identify the divisor and reverse the sign of the constant term. Put this term in the box B. Write the dividend in standard form with exponents in descending order. Put the coefficients of the dividend beside the box on the top row. Be sure to fill in zeros for any missing terms. C. Bring down the first coefficient D. Multiply the first coefficient by the new divisor, identify the result under the next coefficient and add. E. Repeat the steps of multiplying and adding until the remainder is found F. Go backwards from the remainder and assign variables 4 x 8x 11x Ex: x
II. Long Division A. Write both the dividend and divisor in standard form, filling in zeros for any missing terms. B. Count the number of terms in the divisor and move over this many terms. C. What do you have to multiply the first term by to get the dividends first term? D. Multiply this value through the divisor and write it under the like terms in the dividend. E. Subtract (meaning write the negative sign then distribute) F. What do you have to multiply the first term by to get the dividends new 1 st term? G. Continue the process. NOTE: Long division can be used no matter what size the divisor is. 6x 4x + x Ex: x III. Theorems A. Remainder Theorem If polynomial f(x) is divided by x c, then the remainder is f(c). B. Factor Theorem A polynomial f(x) has a linear factor x a if and only if f(a) = 0 In other words: If you get a remainder of 0 by any method, then the number a is a zero and x a is a factor. III. Model Problems 1. 5 4 x x x x x + 5 + 6 + 4 + 9 x +. 6x 19x 6x 4 x. 5 x 1 x 1 4. 6x 0x + x + 8 x
5. 4 x + 4x 5x + x x + x 6. x 4x + 5 x Determine which of the given values are zeros to the given function: 0, f ( x) = 6x 4x + x 5 Determine which of the given values are zeros to the given function: 1, -½ f ( x) = x 4x + x + 4 Find the remainder when x 8x 1x is divided by (x + ) 4 1 Find the remainder when x 11x + 7x is divided by (x 1) ( ) Use the Factor Theorem to determine if x 1 4 factor of 4x + x 6 is a ( ) Use the Factor Theorem to determine if x + 4 a factor of 4 x 4x 8 is
Practice # Divide using Long Division 6x 0x + x + 8 1) x Divide using Synthetic 4 x 8x + 9x + 5 ) x ) 4) x 8 x 4x x + x + 7 x Divide using any method 4 x + 5x x 8 5) x + 6) 4 x + 6x 9x + 5 x 1 7) 5 x 1 x 1 8) x + x 6x x 9) x x x + 5 6 x + x 1 10) 5 4 x x x + 81 81 x + 9 11) 4 5x + 5x + 5 x x 1) 4 x x x x + x + x 1 Determine which of the given values are zeros of the given polynomial. 4 1),, 0, -1; g( x) = x + 6x x 0x 14) 1,½,, -½, f ( x) = 6x + x 1 Find the remainder when f(x) is divided by g(x). f(x) g(x) Remainder 10 8 15) f ( x) = x + x g( x) = x 1 4 16) f ( x) = x 6x + x 1 g( x) = x 17) f ( x) = x x + 5x 4 g( x) = x + 5 4 18) f ( x) = x x + x x + x 8 g( x) = x 10 5 4 19) f ( x) = x x + x 8x 8 g( x) = x 0 Determine if h(x) is a factor of f(x) 5 f ( x) = x 0) 1) h( x) = x 1 f x x x x ( ) = 4 1 h( x) = x + ) f x x x 99 56 ( ) = 14 65 + 51 h( x) = x 1 Lesson #: Real Zeros I can Find all rational zeros of a polynomial function. Apply the Rational Root Theorem. I. Rational Root Theorem n A. All potential rational roots (zeros that are fractions or whole numbers) of the polynomial px +... + q = 0 q come from all factors in the form ±. p 8 Example: List all potential rational roots of 4x x 0x 15 = 0 B. You must include the on all fractions (it doesn t distribute) and reduce all fractions without repeating any.
II. How to find all rational zeros A. Determine the number of zeros (it is equal to the ) B. Graph the function on a graphing calculator. C. Where does the graph cross the x-axis? You can use the graph or the table. D. Using that value, do. Did it really work? If so, write the new equation. E. Can you this equation? If not, continue to do until you can factor it. F. Your answers will not be all whole numbers and so the table will not give you all solutions. III. Model Problems GP 4 Given the equation: x + x 17x 4x + 6 = 0. How many solutions are there? OYO Given the equation: How many solutions are there? 6 5 4 x 4x 5x + 4x + 8x 16x = 0. List all potential rational solutions. List all potential rational solutions Find a whole number zero and do synthetic division using this number. Find a whole number zero and do synthetic division using this number. Find another zero (maybe rational) and reduce the function further. Find another zero (maybe rational) and reduce the function further. Find the remaining zeros. Find the remaining zeros.
Practice # In each question, list all potential rational zeros. Use a calculator and synthetic division to find all real zeros. 1) x x x + = 0 ) x x x + = 0 ) x 5x + x + = 0 4) 4 x x x x + 19 + 1 = 0 5) 5 x x x 5 4 + = 0 6) x + x 7x + x = 0 7) x 4x + x = 0 8) Graph number 7 and state what you notice. How many solutions are there? How many can you find? Why? Lesson #4: Graphing Polynomials I can Write a polynomial function given zeros/factors. Graph a polynomial I. Graphs and writing equations A. If a root is singular, it passes through the x-axis. B. If a root is a double root (what do double roots do best? bounce) then it hits the x-axis and bounces off. Any higher even exponent does the same thing just comes in steeper then flattens out as it approaches the zero. C. If a root is triple or higher odd, then it curves like a cubic equation but comes in steeper then flattens out as it approaches the zero. Examples of the graphs:. D. Writing functions given zeros Steps: 1. Locate the zeros. Put all zeros in the form of (x c) where c is a zero. Multiply the factors 4. Simplify E. Examples Write the simplest polynomial function with the given zeros of -,, 4 Write a polynomial with a scalar of 1 and zeros of 1, 0, and - (DR) II. Basic Polynomial Shapes A. 1 st Degree nd Degree rd Degree B. types: or based on the C. If the is NEGATIVE, the graph is REFLECTED across the x-axis.
III. Properties of Polynomials A. Continuous everywhere (means NO jumps, gaps (Asymptotes), holes, or SHARP corners) B. End behavior tells you the type of of the polynomial. 1. If degree the end behavior of the graph shoots in OPPOSITE directions.. If degree the end behavior of the graph shoots in the SAME direction. C. Examples: For each graph state the 1) end behavior, ) smallest possible degree, and ) sign of leading coefficient. D. Intercepts 1. Every polynomial has exactly 1 y-intercept, which is equal to the, a 0. X-intercepts of a polynomial are where the graph the x-axis. The number of x- intercepts CANNOT exceed the exponent (or degree) of the polynomial. So the number of x-intercepts is LIMITED! E. Multiplicity 1. If is a factor that occurs M times in the complete factorization of a polynomial, then C is called a ZERO WITH MULTIPLICITY M.. Therefore, if m, the exponent is i. ODD, the graph CROSSES the x-axis at c. 11. EVEN, the graph TOUCHES the x-axis at c.
. Examples: Find all zeros of f(x). State the multiplicity of each zero and whether it touches or crosses the x-axis for each zero. ***It is helpful to think, :Does it look cubic, quadratic, or linear at x=c? Practice #4 1-) Determine whether the given graph could possibly be the graph of a polynomial function. 4. The graph of a polynomial function is shown below. List each zero of the polynomial and state whether its multiplicity is even or odd. 5 6) For each complete graph of a polynomial determine: a) Even or Odd degree, b) Is the leading coefficient positive or negative, c) what are the real zeros, and d) what is the smallest possible degree? 7 10) a)list all possible rational roots, b)factor completely using integers, and c)find all the real zeros. Give exact answers.add D) SKETCH THE GRAPH 4 7) f ( x) = x + 5x 8x 15 8) f ( x) = x 1x 1x 6 4 9) f ( x) = x + x x + 4 10) ( ) f x = 1x + 5x 1x 9x 6 11 16) Determine the following for each polynomial: a)state the total number of zeros, b)state the end behavior, c)the consecutive integers that the real roots are located between, and d)graph. 17) Find the value of k so that x + is a factor of x x kx + + 18) Find the value of k so that x 1 is a factor of
Lesson #5: Real Life Data I can Model real life data with polynomials
Lesson #6: Fundamental Theorem of Algebra I can Find all zeros of a polynomial equation. Write a polynomial equation given some zeros. Determine how many positive, negative, and imaginary zeros are possible. Determine the least upper bound and greatest lower bound. I. Theorems A. Fundamental Theorem of Algebra: The number of solutions in a polynomial equation with rational coefficients is equal to the degree of the polynomial when including all complex solutions and solutions of multiplicity. What does this mean? You are responsible for finding all solutions even if they are imaginary, if they are double roots are larger, and if they are real. Whatever the highest exponent is, that is how many you must account for. B. Conjugate Root Theorem: All complex zeros (imaginaries and radical) come in conjugate pairs. What does this mean? If you know + i is a zero then so is. If then so is. C. To write a polynomial equation 1. Look to see if there are any imaginary or radical solutions. If there are, include its conjugate.. List all of the factors (recall you take the zeros and put in front of them).. Sum and product: The complex factors are more difficult to FOIL so you can use the idea that the complex factors will always make x bx + c where the zeros multiply to make c and add to make b (NOTICE you must the on b because of the negative in front of this term!!!) 4. FOIL the remaining factors and set to p(x). II. How to find all zeros A. Use the rational root theorem ( over ) to make a list of potential answers. B. the function C. Determine which values appear to be. D. Do using these zeros to get a equation. E. Continue to reduce until you can factor or until you got it down to a equation. You can now use the quadratic formula: III. Least Upper Bound & Greatest Lower Bound A. Key idea: All zeros are between the LUB and GLB so if you can determine these values, you don t need to check numbers further away. Both use synthetic division. B. Upper Bound To check if a positive value is an upper bound, its row would contain only nonnegative values when doing synthetic division. C. Lower Bound To check if a negative value is a lower bound, the row when doing synthetic division would have signs that alternate with 0 being considered either sign. IV. Descartes s Rule of Signs A. The number of positive real zeros of f(x) is either equal to the number of sign changes in the function or less by the number by an even integer. Example: f ( x) = x 5x + 6x 4 There are sign changes so there could be + solutions or + solution. It is impossible to have or positive solutions out of the total solutions. B. Substitute (-x): The number of negative real zeros of f(x) is either equal to the number of sign changes in f(-x) or less by the number by an even integer.
Example: From the example above, f ( x) = x 5x + 6x 4 becomes sign changes so there is - solutions. C. Build a chart to represent your possibilities. f ( x) = x 5x 6x 4There are Positive Zeros Negative Zeros Imaginary Zeros V. Model Problems GP If you know that + i is a zero of x 5x + 7x = 0, find the other zeros. OYO If you know that - i is a zero of x 8x + x 0 = 0, find the other zeros Find all zeros of: 4 f ( x) = x 5x + 4x + x 8 Find all zeros of: g( x) = x 8 Write the polynomial equation with known zeros of, 1, and 4 i Write the polynomial equation with known zeros of 0,,i
Write the polynomial equation with known zeros of, + Write the polynomial equation with known zeros of 1,1 5 Verify that 1 is an upper bound for f ( x) = 6x 4x + x Verify that is an upper bound for f ( x) = x 5x + 6x 4but 1 is not. Verify that - is a lower bound for 4 f ( x) = x 4x 6x 16 Verify that -1 is a lower bound for 4 f ( x) = x 4x 5 Verify that is an upper bound and -4 is a lower bound to 4 f ( x) = x 8x + Verify that 4 is an upper bound and - is a lower bound to f ( x) = x x 1x + 8 Make a chart of Descartes s Rule of Signs for: 4 h( x) = x + x Positive Negative Imaginary Make a chart of Descartes s Rule of Signs for: 4 h( x) = x x + Positive Negative Imaginary Make a chart of Descartes s Rule of Signs for: f ( x) = 4x x + x 1 Positive Negative Imaginary Make a chart of Descartes s Rule of Signs for: f ( x) = 5x + x x + 5 Positive Negative Imaginary
Practice #6 1 6: Find all zeros for each function. 4 1) f ( x) = x + 7 ) f ( x) = x 16 ) 4 f ( x) = x x 10 4) f x x x 4 ( ) = 7 4 5) f ( x) = x x x 6) f x x x x x 4 ( ) = + 5 8 + 4 7 & 8: Find the remaining zeros if one zero is given to be the number i. 4 7) f ( x) = x + x + x 8) f ( x) = x x 5x x 6 Write the polynomial equation given the known zeros. 9) 1,7, 4 10),0,5 11) i 1) 1+ i 1), 4 i 14) double root, + i 15) Verify that is an upper bound to 16) Verify that - is a lower bound to 6 f ( x) = x + x 7x x 6 f ( x) = x + x 7x x 17) Verify that 6 is an upper bound to 18) Verify that - is a lower bound to 7 6 5 4 f ( x) = x 6x + 9x + 7x 8x + x 6x + 0 For each function below, make a chart to represent Descartes s Rule of Signs. 6 19) f ( x) = x + x 7x x 7 6 5 4 0) f ( x) = x 6x + 9x + 7x 8x + x 6x + 0 1) f ( x) = x + x + x + ) h( x) = 4x 8x + 9) ) 4 f ( x) = x x + x x 6 4) g x x x ( ) = + 1 MIXED PRACTICE WILL BE SEPERATE