U7 - Polynomials. Polynomial Functions - In this first section, we will focus on polynomial functions that have a higher degree than 2.

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1 U7 - Polynomials Name 1 Polynomial Functions - In this first section, we will focus on polynomial functions that have a higher degree than 2. - A one-variable is an expression that involves, at most, the operations of additions, subtraction, and multiplication. - The terms of the polynomial are usually written in order, starting with the highest power of x, and going down from left to right. - The highest power of the polynomial is the of the polynomial. - Also, the number in front of each x is known as the. Example 1: f ( x) = 7x x x + 7 Degree of this function? Coefficients? - 7 is known as the term, because it doesn t have a x term to go with. Example 2: y = 2( x + 2) ( x + 5) Degree of the function? Coefficients? - The degree of the polynomial tells you things like what type of it is and how many it ll have. - The roots, or, of a function are the solutions to the function when you set the function equal to 0. For instance, use what you know to answer the following questions. a. What is the maximum number of roots a polynomial of degree 3 can have? b. What do you think is the maximum number of roots a polynomial of degree n can have?

2 2 c. How many roots does x 2 + 6x + 9 have based on the degree of the function? d. When you solve for the zeros of the previous function, how many do you get and why? - When the y-values of a graph increase as the x-values increase, the graph has orientation. When the y-values of a graph decrease as the x-values increase, the graph has orientation. 1. For each polynomial function shown below, state the minimum degree its equation could have and how its oriented? i. Degree? ii. Degree? iii. Degree? iv. Degree? - A is a number or quantity that when multiplied with another produces a given number or expression. - Each factor represents a change in on the graph. If a factor appears more than once, it is. Reoccurring factors do not continue to have an affect on the graph, they only affect the graph once. 2 - If you have a factor that is squared, such as x a, you have what is called a root. The affect on the graph is that where x = a the graph will not cross the x-axis, just touch it before going back in the direction from which it came. - If you have a factor that is cubed, such as x a, you have what is called a root. The affect on this graph is that the graph will have an inflection point where x = a, meaning it will switch between opening up to opening down or vice versa How do the shapes of graphs of y = (x 2) 3 and y = (x + 1) 5 with repeated factors differ from the shapes of graphs of equations that have three or five factors that are different from one another?

3 3. What can you say about the graphs of polynomial functions with an even degree compared to the graphs of polynomial functions with an odd degree? 3 4. Without using a calculator, sketch rough graphs of the following functions. = x( x +1) ( x 3) f ( x) = ( x 1) 2 ( x + 2) ( x 4) f ( x) = ( x + 2) 3 ( x 4) a. f x b. c. = ( x 1) 2 ( x + 2) ( x 4) f ( x) = a( x 1) 2 ( x + 2) ( x 4) - If I were to take f x from above and alter like. How did a change my original graph? 5. THE COUNTY FAIR COASTER RIDE You have been hired by a theme park to find the exact equation to represent the roller-coaster track on the graph. The numbers along the x-axis are in hundreds of feet. At 250 feet, the track will be 20 feet below the surface. This gives the point (2.5, 0.2). a. What minimum degree polynomial represents the portion of the roller coaster represented by the graph, and what are its roots? Be specific. Degree? Roots? b. Find an exact equation for the polynomial that will generate the curve of the track. c. What is the deepest point of the roller coaster's tunnel?

4 6. Some polynomials have a stretch factor, just like the a in parabolas and other parent functions. Write an exact equation, including the stretch factor, for each graph below. 4 a. Equation: b. Equation: Imaginary Numbers and Complex Roots - Not all quadratic equations have real-number solutions. For instance, x 2 = 1 has no real-number solutions because the square of any real number x is never. To overcome this problem, mathematicians decided to use an expanded number system using the unit i, which is defined as i = 1, so i 2 =. The imaginary unit i can be used to write the square root of negative number. - Here is the pattern for i : i = 1 i 2 = 1 i 3 = 1 i 4 = 1 i = i i 2 = 1 i 3 = i i 4 = 1 - Use the definition of i to rewrite each of the following expressions. ( 3i) = ( 2i) 2 ( 5i) = 25 = a. 4 = b. 2i c. d.

5 - A number written in standard form is a number where a and b are real numbers. The number a is the part of the complex number, and the number bi is the part. 5 Solve the following equations. a. 3x = 26 b. x 2 = 9 c. 2x = 13 d. x 1 2 = 7 - When plotting a complex number on a graph, you use the plane instead of the standard (x, y) plane. The x-axis becomes the axis, and the y-axis becomes the axis. Plot the complex numbers on the complex plane. e. 2 3i f. 3+ 2i g. 4i - Just like real numbers, the same order of operations apply to numbers. Therefore, when adding and subtracting, you only combine. Write the expression as a complex number in standard form. + ( 3+ 2i) ( 7 5i) ( 1 5i) 6 ( 2 + 9i) + ( 8 + 4i) h. 4 i i. j. ( 7 4i) ( 1+ 2i) ( 6 + 3i) ( 6 3i) k. 5i 2 + i l. m.

6 6 - Notice from the previous problem, you have two factors in the form a + bi and a bi. These are complex, and the product of these numbers is always a number. You can use complex conjugates to write the of two complex numbers in standard form. Write in standard form i n. o. 1 2i 3 4i 1+ i - The absolute value of any number, by definition, is its. Find the absolute value of each complex number. p. 3+ 4i q. 2i r. 1+ 5i - Which number is the farthest from the origin in the complex plane? Practice Solve the equation. 1. x 2 = 4 2. x 2 = x 2 = x = x = 3 6. x 2 4 = r = 5r x 2 1 = 7x 2 9. y 2 2 = 16

7 ( u + 5) 2 = ( x + 3)2 = 7 9 w = 0 Plot the numbers in the same complex plane i i 15. 4i i i i i Write the expression as a complex number in standard form. + ( 7 + i) ( 6 + 2i) + ( 5 i) ( 4 + 7i) + ( 4 7i) i ( 9 3i) ( 8 + 5i) ( 1+ 2i) ( 2 6i) ( i) i ( i) ( i) ( 25 6i) i + ( 8 2i) ( 5 9i) i ( i) + 30i i( 3+ i) 4i( 6 i) i

8 ( 5 + i) ( 8 + i) ( 1+ 2i) ( 11 i) i 4 + 7i ( 9 6i) ( 7 + 5i) ( 7 5i) ( 3+10i) i i i 2i 1 i 5 3i 3+ i i 3 i 2 + 5i 5 + 2i 7 + 6i i 10 i 6 i i 2 Find the absolute value of the complex number i 49. 2i i i i i i i 5

9 Long Division of Polynomials 9 - When you have a polynomial with a degree higher than, it is hard to by our usual methods. Therefore, we need to use a different approach that will work regardless of the degree of the polynomial. First, we will use to factor. - What are the components of a division problem? Divide the polynomial 6x 3 19x 2 +16x 4 by x 2, and use the result to factor the polynomial completely. Divide the polynomial 3x 2 +19x + 28 by x + 4, and use the result to factor the polynomial completely. - Sometimes when performing, you end up with a remainder. So, how do we account for this?

10 Use long division to divide. 10 x + 3 5x 2 17x x 2 +10x x 4 4x + 5 6x 3 16x 2 +17x x 3 7x 2 11x x 2 x +1 x x 2 + 3x x 1 x 3 x 3 + 9x + 6x 4 x x 3 2x x

11 Synthetic Division 11 - There is a nice shortcut for long division polynomials by divisors of the form x k. Use synthetic division to divide. x 5 5x 3 +18x 2 + 7x x 3 17x 2 +15x x + 3 x 3 2x 3 +14x 2 20x x 3 + 7x 2 x x + 6 x + 3 5x 3 + 8x 2 x x 4 10x 2 2x x + 2

12 Remainder Theorem 12 - This is a simple, easy-to-use way of evaluating a polynomial function. If a polynomial f x is divided by x k, then my remainder and solution would f k. Use the Remainder Theorem to evaluate f ( x) = 3x 3 + 8x 2 + 5x 7 when x = 2 = 4x 3 +10x 2 3x 8 Use the Remainder Theorem to find each function value given f x. 1 a. f ( 1) b. f ( 4) c. f d. 2 f ( 3) Rational Zero Test - Lets remind ourselves what it means to have a rational number. A rational number is any number that can be written as a. We are going to use this concept to find real rational zeros of functions. - When your leading coefficient is 1, all of your potential roots are the factors of your. Find the rational zeros. = x 3 + x +1 f ( x) = x 3 5x 2 + 2x f x 2.

13 = x 3 + 2x 2 + 6x 4 f ( x) = x 3 3x 2 + 2x 6 3. f x When you have factored a polynomial from scratch with no initial roots given, then factoring a perfect cube will be a breeze. Here are the basic forms. a. a 3 + b 3 b. a 3 b 3 - Use the basic forms above to factor the following. 5. x x x x x x x x x x x x