Pre-Calculus Notes: Chapter 4 - Polynomial and Rational Functions. Polynomial in One Variable,x. Degree. Leading Coefficient. Polynomial Function

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1 Name: Pre-Calculus Notes: Chapter 4 - Polynomial and Rational Functions Section 1 Polynomial Functions Polynomial in One Variable,x Degree Leading Coefficient Polynomial Function Zeros / Solutions / Roots / (x-intercepts) Example1 Consider the polynomial function f(x) = 3x 4 x 3 + x + x 1. a. State the degree and leading coefficient of the polynomial. b. Determine whether - is a zero of f(x). Imaginary Number Complex Numbers Fundamental Theroem of Algebra 1

2 Corollary to the Fundamental Theorem of Algebra The general shapes of the graphs of polynomial functions with positive leading coefficients and degree greater than 0 are shown below: Since the x-axis only represents real numbers, imaginary roots cannot be determined by using a graph. What must be true about the number of x-intercepts for a polynomial with an even degree? What must be true about the number of x-intercepts for a polynomial with an odd degree? If you know the roots of a polynomial equation, you can use the corollary to the Fundamental Theorem of Algebra to find the polynomial equation. That is, if a and b are roots of the equation, the equation must be (x a)(x b) = 0 Example Write a polynomial equation of least degree with roots, 3i, and -3i. Does the equation have an odd or even degree? How many times does the graph cross the x-axis?

3 Example 3 State the number of complex roots of the equation 3x 3 3x + 4x 4 = 0. Then find the roots and graph the related function. Example 4 State the number of complex roots of the equation 9x 4 35x 4 = 0 and then find the roots. Example 5 When a golf ball is hit from a tree with a velocity of 160 ft/s at an angle of 45 o with respect to the ground x (horizontal), the height (in feet) of the ball above the ground is given by h( x) = x, where x is the 800 horizontal distance from the tee. a. How far from the tee does the ball strike the ground? b. Verify your answer using a graph. 3

4 Section Quadratic Equations There are four methods we can use to solve quadratic equations: graphing, factoring, completing the square, and the quadratic formula. Quadratic Formula: x = b ± b 4ac a To determine how many solutions and the type of solutions a quadratic function will yield, we refer to the discriminant: b 4ac. Example 1 Solve 3x + x = 0 Example Solve by completing the square. a. x 6x 7 = 0 b. x 6x + 13 = 0 4

5 Example 3 Find the discriminant of x + x = 0 and describe the nature of the roots of the equation. Then solve the equation by using the Quadratic Formula. Example 4 A late-night talk show host organized a filming stunt from a 00-ft-tall building in a city s downtown. She launched a cantaloupe from the tower s roof at an upward initial velocity of 70 ft/s. A film crew recorded the fruit s messy fall into the roped-off area below. The height of the cantaloupe is given by h(t) = 70t 16t + 00 where t is the number of seconds since the fruit was launched. How long will it take the cantaloupe to hit the ground? Conjugates Complex Conjugates Theorem Example 5 Solve x 4x = -15 Example 6 Solve each equation using substitution. a. x 6 + 7x 3 8 = 0 b. y 10 5y 6 + 4y = 0 5

6 c. e x e x 6 = 0 d. e 4x = 13e x 36 e. (5 x 4) = 5 x + 3 f. 1 x + 8 = 6 x Section 3 The Remainder and Factor Theorems Polynomial Division: Example 1 Divide f(a) = a + 3a 8 by a. Method 1 Good old-fashioned long division a a + 3a 8 Method The box Method 3 Synthetic division (can only be used if the divisor can be written in the form x r) 6

7 Remainder Theorem Example Divide y 6 + 4y 4 + 3y + y by y + using synthetic division. Factor Theorem Example 3 Use the Remainder Theorem to find the remainder when x 3 x 5x 3 is divided by x 3. State whether the binomial is a factor of the polynomial. Explain. Example 4 A factor of x 3 3x + x is x 1. Determine the remaining factors. Example 5 Determine the binomial factors of x 3 x 13x 10. 7

8 Example 6 Find the value of j so that the remainder of ( 3 + 6x jx 8) ( x ) x is 0. Section 5 Locating Zeros of a Polynomial Function Locating Intervals Where the Polynomial is Positive or Negative The Location Principle Example 1 Determine between which consecutive integers the real zeros of f(x) = 1x 3 0x x + 6 are located. Example Approximate the real zeros of f(x) = -3x x 3 18x + 5 to the nearest tenth. State the interval(s) on which f(x) is positive and negative. Example 3 Approximate the real zeros of f(x) = 1x 3 19x x + 6 to the nearest tenth. State the interval(s) on which f(x) is positive and negative. Example 4 The Toaster Treats Company uses a box with a square bottom to package its product. The height of the box is 3 inches more than the length of the bottom. Find the dimensions of the box if the volume is 4 in 3. 8

9 Section 6 Rational Equations and Partial Fractions Rational Equation Example 1 Solve for x. Check for extraneous solutions. a. 5 3 = x x 1 b. 1+ x 8 16 = 1 x 4 c. x 0 = x + 4 x 1 x + 3x 4 d. t = t + 1 t 3 t 16 t 3 9

10 Example x x 3 ( )( x 1 ) Solve ( )( x 4) < 0. Example 3 x + ( )( x 3 ) Solve ( x 1)( x + 1) 0 Example 4 5 Solve + 3 a 6 a > 3 4 Interval/ Test Point Check Yes/No 10

11 Example Solve + > b + 1 b Interval/ Test Point Check Yes/No Section 8 Modeling Real-World Data with Polynomial Functions Example 1 Determine the type of polynomial function that could be used to represent the data in each scatter plot. a. b. c. 11

12 Example Use a graphing calculator to write a polynomial function to model the set of data. Example 3 Listed below are the number of fat grams and the corresponding Calories for single servings of several convenience items. a. What polynomial function could be used to model these data? b. Use the model to predict the number of Calories for a similar food item having 10 grams of fat per serving. c. Use the model to predict the number of fat grams for a similar food item having 450 Calories per serving. 1