1. Chapter 4: Polynomial Functions

2. Student will be able to Determine roots of polynomial equations. Apply the fundamental Theorem of Algebra A polynomial in one variable is a polynomial that contains only one variable (usually x). The degree of a polynomial in one variable is the greatest exponent of its variable. The coefficient of the variable with the greatest exponent is called the leading coefficient. The zeros of a polynomial function are the values that make the polynomial equal 0. These values are solutions to the polynomial equations and can also be called the root of the equation.

3. The Fundamental Theorem of Algebra says that every polynomial with degree greater than zero has at least one root in the set of complex numbers (real, imaginary, fractions, whole, integers, etc.) Every polynomial with degree n, can be written as n factors. Examples: Since the quadratic has a degree of 2, it will have 2 factors, (x+3) and (x+2) Since there is a degree of 3, it will have three factors, (x+2)(x+3)(x+4) If you have a factor (x+a) for a polynomial, then you have a root/solution/zero of x = -a. When a root is imaginary, it doesn’t show on the graph. If you have one imaginary root, you also have its conjugate.

4. Write a polynomial equation of least degree with roots 2, 3i, and -3i. Does the equation have an even or odd degree? How many times does the graph of the related function cross the x-axis? Example: If you have the following factors: (x+3) and (x+4i), you’ll know the following: 1.If you have a factor of (x+4i), then you must have a factor of (x-4i). 2.You have three factors, but the graph will only cross the axis one time, at x = -3, because it is the only real root.

6. Sec 2: Quadratic Equations The student will be able to: Solve quadratic equations Use the discriminant to describe the roots of quadratic equations. Several methods are used to solve quadratic equations: Completing the square Factoring Graphing Quadratic Formula The quadratic formula and completing the square methods will provide you will all possible complex roots, including imaginary and irrational. Factoring will only work for integers, imaginary numbers, and fractions.

7. Quadratic Formula for quadratics of the form Ax +By = C: For the example listed above, a = 3, b = 7, and c =7. Solve using quadratic formula:

8. Completing the square is a little more complicated, but as long as you do each step, you should be able to do it. Here are your steps: 1.If the leading coefficient is other than 1, divide through the entire equation by that number. 2.Subtract the constant to the other side of the equal sign. 3.Take half the middle, square it, and add it to both sides of the equation. 4.Rewrite the left side as a binomial squared and simplify the right side if possible. 5.Square root both sides and solve Solve using completing the square.

9. Section 3: The Remainder and Factor Theorems The student will be able to find the factors of polynomials using the Remainder and Factor Theorems. The Remainder Theorem states that if a polynomial is divided by x-r, then the remainder after division is the same as f(r). Synthetic Division is a shortcut for dividing a polynomial by a binomial factor.

10. Section 4: Rational Root Theorem The student will be able to identify all possible rational roots of a polynomial equation by using the Rational Root Theorem. Rational Root theorem says you can find the possible rational roots of a polynomial by finding State the possible rational zeros for each function. Then find all the zeros.

11. Section 8: Modeling Real-World Data with Polynomial Functions Copy the purple and white box on page 258. All of it, including the graphs. Using a graphing calculator, write a polynomial function to model the set of data. x-0.500.511.522.533.54 f(x)-10-6.4-5-5.1-6-6.9-7-5.6-24.615 1. x-2-1.5-0.500.51.01.522.53 f(x)-22-7.903.54.12.92.12.55.814.128 2.

12. Listed below are the number of fat grams and the corresponding Calories for single servings of several convenience items. ItemFat(g)Calories 2% milk5120 Tortilla chips8160 Peanut Butter16225 Popcorn9150 Coffee cake11210 Danish11260 Chicken nuggets16280 Yogurt280 Pizza (frozen)16390 Sugar cookies4.5150 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.