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Odd Degree Even Unit 5 Higher Order Polynomials Name: Polynomial Vocabulary: Polynomial Characteristics of Polynomials and their Graphs of the polynomial - highest power, determines the total number of Leading - coefficient of and variable with highest power Leading - coefficient of the leading term End Behavior: what happens to the value of the as x values approach and. Sign of Leading Coefficient Positive Negative as x, y as x, y as x, y as x, y as x, y as x, y as x, y as x, y Forms of Polynomials Equations/Functions: Separate Terms: Factored: f(x) = (x 1)(x + 4)(x 2 + 9) Characteristics of Graphs 1. Turns: points where the graph changes from (walking uphill) to (walking downhill) and vice versa happen at maxs and mins always less than the 2. X-intercepts where the graph OR the x-axis Double x-intercepts same solution when solving double factor graph only, does not x-axis Imaginary x-intercepts always in X-intercept rules: Even degree o 0 x-intercepts OR o Any number the degree 3. Y-intercept evaluate the polynomial at x = 0 Odd degree: o 1 x-intercept OR o Any number the degree 1

Variations of Common Polynomials Linear (y = mx + b) (Degree 1) Quadratic (y = x 2 ) (Degree 2) One real root Two real Two imaginary One real (double root) y = y = y = y = Cubic (y = x 3 ) (Degree 3) Three real roots: One real / two imaginary roots: One real root (triple root): y = y = y = Quartic (y = x 4 ) (Degree 4) Four real roots: Two real / two imaginary: Four imaginary roots: One real (quadruple root): y = y = y = y = Quintic (y = x 5 ) (degree 5) Five real roots: 3 real roots/2 imag: 1 real (4 imag): y = y = y = Without using a graphing calculator, match the equations with the correct graph. f(x) = 3x 3 8x 2 + 4x g(x) = x 3 + 2x 2 3 h(x) = 1 3 x4 2x 2 + 1 k(x) = 3x 4 + 5x 3 4x + 1 Roots: Roots: Roots: Roots: # of Turns: # of Turns: # of Turns: # of Turns: y-int: y-int: y-int: y-int: End behavior: End behavior: End behavior: End behavior: 2

Finding Roots/Solving Polynomials More Polynomial Vocabulary: f(x) = 3x 2 + 4x 7 x = 1 is a of f(x). x = 1 is a of f(x) because f(1) = 0. (x 1) is a of f(x). (1,0) is a of the graph of f(x). To solve/find the roots of a polynomial: 1. Take out f(x) and put in 0 OR move all terms to one side and get 0 on the other 2. Use one of the methods based on the degree of the polynomial. 3. Write answers as ordered pairs: Methods for Degree 2 ONLY 1. Quadratic Formula 2. Taking the of Both Sides 3. Difference of Two Squares Methods for Degree 3 ONLY 1. Sum of Cubes 2. Difference of Cubes Methods for Degree 2 and 4 Trinomials 1. Degree 2 a =1 2. Degree 4 a =1 a 1 a 1 Methods for ALL Degrees 1. GCF 2. Factor by Grouping 3

Graphing Polynomials 1. f(x) = 2x 3 6x 2 Degree = EB: Poss. # of turns = x-intercepts = Actual = y-int = 2. f(x) = x 4 + 4x 2 Degree = EB: Poss. # of turns = Actual = x-intercepts = 3. f(x) = x 3 2x 2 x + 2 Degree = EB: Poss. # of turns = Actual = x-intercepts = y-int = y-int = 4

4. f(x) = x 4 6x 2 + 9 Degree = EB: Poss. # of turns = Actual = x-intercepts = 5. f(x) = 2x 5 + 4x 4 3x 3 6x 2 Degree = EB: Poss. # of turns = Actual = x-intercepts = y-int = y-int = Dividing Polynomials Method 1 Long Division Example 1: (x 3 3x 2 + x 8) (x 1) Example 2: ( 3x 3 + 4x 1) (x 1) Example 3: (3x 3 2x 2 + 4x 1) (x 2 + 1) 5

Method 2 - Synthetic Division: process of dividing a polynomial by Steps: Example 1: Divide f(x) = x 3 + 2x 2 6x 9 by x + 3 1. Arrange the terms of the polynomial in descending powers of x. Insert zeros for any missing powers of x 2. Write the constant r of the divisor x k. 3. Bring down the first coefficient. 4. Multiply the first coefficient by k. Then write the product under the next coefficient. Add. 5. Multiply the sum by k. Then write the product under the next coefficient. Add. Repeat Step 5 for all coefficients in the dividend 6. Bottom line are the coefficients for each term. Start one degree less than the dividend. Last number is the remainder. Example 2: Divide f(x) = x 4 + x 3 5x 2 + x 6 by x 2 The Remainder Theorem: If a polynomial f(x) is divided by x k, the remainder (r) is the value of f(k). Ex. If f(x) = x 3 + 3x 2 7x 6 and f(5) = 159 then the remainder of x3 +3x 2 7x 6 is 159. Application: Determine the remainder when the polynomial f(x) = x 3 + 3x 2 7x 6 is divided by x + 4. x 5 The Factor Theorem: A polynomial f(x) has a factor (x k) if and only if f(k) = 0. Ex. Given f(x) = x 3 + 3x 2 7x 6, since f(2) = 0 the remainder (after synthetic division) is zero therefore (x 2) is a factor and x = 2 is a zero or root. Application: Determine if the values are roots/zeros of the polynomial f(x) = x 3 2x 2 9x + 18. a. x = 2 b. x = 4 6

Rational Root Theorem Example 1: f(x) = 2x 4 + x 3 14x 2 19x 6 Finding All Roots to a Polynomial 1. List all possible p values (factors of p = constant) 2. List all possible q values (factors of q = leading coefficient) 3. List all possible roots p q 4. Test the roots using the remainder theorem. 5. Use synthetic division for the roots that work 6. Try to factor the depressed polynomial, or complete the rational root test with the polynomial Example 2: f(x) = 2x 4 9x 2 + 7 Example 3: f(x) = 3x 4 10x 3 24x 2 6x + 5 7

Analyzing Graphs of Polynomials: Approximating a Relative Minimum and Relative Maximum: 1. 2 3 2 f ( x) 3x 4x 2 2. f ( x) x 3x 2 3. f ( x) 3x 6x 4 2 Increasing and Decreasing Functions: 4. f ( x) 3 x 5. 3 f ( x) x 3x 6. t1, t 0 f ( t) 1,0 t 2 t 3, t 2 Increasing: Increasing: Increasing: Decreasing: Decreasing: Decreasing: Calculator Work: Sketch the graph of the following functions. 3 1. f ( x ) 2x 5x 6 2. f x x x 2 ( ) 4 1 Max/Min Calc Steps Type graph in y = Hit Zoom #6 Adjust Window to see a closer view (if needed) Hit Graph Hit 2 nd Calc #4 Max or #3 Min Move cursor Left Bound, Enter Move cursor Right Bound, Enter Move cursor to the middle(guess), Enter Zeros: Relative Maximums: Relative Minimums: Increasing: Zeros: Relative Maximums: Relative Minimums: Increasing: Zeros Steps Type graph in y = Hit Zoom #6 Adjust Window to see a closer view (if needed) Hit Graph Hit 2 nd Calc #2 (Zero) Move cursor Left Bound, Enter Move cursor Right Bound, Enter Move cursor to the middle(guess), Enter Decreasing: Decreasing: 8

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