39 Algebra 2 Honors Chapter 2 Notes: Polynomials and Polynomial Functions Section 2.1: Use Properties of Exponents Evaluate each expression (3 4 ) 2 ( 5 8 ) 3 ( 2) 3 ( 2) 9 ( a2 3 ( y 2 ) 5 y 2 y 12 rs b 3) (rs 1 ) 3 More Challenge: ( a2 b a 3 b 2) 1 2mn 2 (3m 2 n) 2 12m 3 n 4
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41 Section: 2.3 Add, Subtract, and Multiply Polynomials Examples: (x 4 + 2x 3 + 8) + (2x 4 9) (x + 2)(5x 2 + 3x 1) (9x 4 12x 3 + x 2 8) (3x 4 12x 3 x) (x 2)(x 1)(x + 3)
42 Special Product Patterns 1) Sum and Difference 2 2 ( a b)( a b) a b 2) Square of a Binomial 2 2 2 ( a b) a 2ab b 2 2 2 ( a b) a 2ab b 3) Cube of a Binomial ( a b) a 3a b 3ab b 3 3 2 2 3 ( a b) a 3a b 3ab b Examples: 3 3 2 2 3 2 3 (3x 2)(3x 2) (5a 2) ( ab 2)
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44 Section 2.2: Evaluate and Graph Polynomial Functions Polynomial Function: A function where a variable x is raised to a nonnegative integer power. The domain of any polynomial is the set of all real numbers. n is the degree of a polynomial is the highest power of x in the polynomial. The coefficient of the highest degree term a n is called the leading coefficient. The term in a polynomial with no variable a 0 is called the constant term and is a coefficient of itself. Example:
What are examples of functions that are not polynomial functions? 45
46 End Behavior: There are four scenarios: 1 2 2 Applying these principles to polynomials in standard form ( ) n n f x a x a x a x n... a x a x a. The end behavior of the polynomial depends on the degree, n, of the polynomial. n n 1 n 2 2 1 0
47 Section: 2.4 Factor and Solve Polynomial Equations Recall how to Factor Quadratic Equations Factor Polynomial in Quadratic Form: 25x 4 36 a 2 b 2 8ab 3 + 16b 4 Factor by grouping: x 2 y 2 3x 2 4y 2 + 12
48 Solving Polynomial Equations by Factoring: Solve: 2x 5 + 24x = 14x 3 You are building a rectangular bin to hold mulch for your garden. The bin will hold 162 ft 3 of mulch. The dimensions of the bin are x ft. by 5x 6 ft. by 5x 9 ft. How tall will the bin be?
49 Graphing Polynomials in Factored Form Sketch the graph: g(x) = (x 1) 3 (x + 2)(x 3) f(x) = (x + 3) 2 (x + 1)(x 2) 3 f(x) = x 5 + 4x 3
50 2.5 Apply the Remainder and Factor Theorems Dividing Polynomials When you divide a polynomial f( x) by a divisor dx ( ), you get a quotient polynomial qx ( ) and f ( x) r( x) a remainder rx ( ). We must write this as qx ( ). d( x) d( x) Method 1) Using Long Division: Divide 4 2 y y y 2 5 by y 2 y 1 Divide 2 8 by x 1 3 2 x x x
51 Let f x x x x 3 2 ( ) 3 2 2 5 1. Use long division to divide f( x ) by x 2 What is the Quotient? What is the Remainder? 2. Use Synthetic Substitution to evaluate f (2). How is f (2) related to the remainder?. What do you notice about the other constants in the last row of the synthetic substitution? The remainder theorem says that when a polynomial f(x) is divided by a linear (1 st degree) polynomial, x c and you solve it for x so it is in the form x = c, if you evaluate f(c), the answer will be your remainder. Example:
52 The factor theorem is an extension of the remainder theorem. Recall that the remainder theorem states that if you evaluate a polynomial with the c-value found by setting the linear x + c = 0, the result will be the remainder of the polynomial had we done either long or synthetic division. The factor theorem extends that by saying if the remainder theorem results in 0, the linear x + c must be a factor of the polynomial. Example: Determine whether: a) x + 1 is a factor of f(x) = x 4 5x 2 + 6x 1 b) x 2 is a factor of f(x) = x 3 3x 2 + 4 The remainder is. This means that x 2 is a factor of f(x) = x 3 3x 2 + 4Therefore you can write the result as:
53 Example: How many zeros does the polynomial have? f(x) = (x 3)(x 1) 2 (x + 2) 3 Factoring Polynomials Factor: 3 2 f ( x) 3x 13x 2x 8 given that f ( 4) 0 Solving Polynomials (which also means Finding the ) One zero of 3 2 f ( x) x 6x 3x 10 is x 5. Find the other zeros of the function. f(x) = x 5 4x 4 7x 3 + 14x 2 44x + 120 if 2, 5, and -3 are factors. Find the other zeros of the function.
54 Using Polynomial Division in Real Life A company that manufactures CD-ROM drives would like to increase its production. The demand function for 2 the drives is p 75 3x, where p is the price the company charges per unit when the company produces x million units. It costs the company $25 to produce each drive. a) Write an equation giving the company s profit as a function of the number of CD-ROM drives it manufactures. b) The company currently manufactures 2 million CD-ROM drives and makes a profit of $76,000,000. At what other level of production would the company also make $76,000,000?
55 2.6 Finding Rational Zeros We call the list of all p possible or potential rational zeros. q Find the rational zeros of 3 2 f ( x) x 4x 11x 30. List the possible rational zeros: Test (Verify zero using the Remainder Theorem) Factor
56 Find the all real zeros of: 4 3 2 f ( x) 15x 68x 7x 24x 4. Solving Polynomial Equations in Real Life A rectangular column of cement is to have a volume of 20.25 than half the height of the column. What should the dimensions of the column be? 3 ft. The base is to be square, with sides 3 ft. less A company that makes salsa wants to change the size of the cylindrical salsa cans. The radius of the new can will be 5 cm. less than the height. The container will hold 144 new container? 3 cm of salsa. What are the dimensions of the
57 2.7 Finding All Zeros of Polynomial Function Use Zeros to write a polynomial function 5 4 2 Example 1: Find all the zeros of f ( x) x 2x 8x 13x 6 Example 2: Write a polynomial function f(x) of least degree that has real coefficients, a leading coefficient of 1, and 2 and 1 + i as zeros.
58 Example 3: Write a polynomial function f(x) of least degree that has real coefficients, a leading coefficient of 2, and 5 and 3i are zeros. 2.8 Analyzing Graphs of Polynomial Functions -Using the Graphing Calculator 1) Approximate Zeros of a Polynomial Function. [2nd][TRACE][zero] 2) Find Maximum and Minimum Points of a Polynomial Function. [2nd][TRACE][maximum or minimum] 3) Find a Polynomial Model that fits a given set of data. (Cubic, Quartic Regression) and make predictions. [STAT, EDIT, input data in L 1 and L 2, STAT, CALC, CubicReg or QuartReg, VARS, Y-VARS, Function Y 1 ] The graph of a function has ups and down or peaks and valleys. A peak is known as a maximum (plural - maxima) and a valley is termed as a minimum (plural - minima) of given function. There may be more than one maximum or minimum in a function. The maxima and minima are collectively known as extrema (whose singular is extremum) that are said to be the largest and smallest values undertaken by given function at some point either in certain neighborhood (relative or local extrema) or over the domain of the function (absolute or global extrema).
59 How many turning points does a polynomial have? Never more than the, we say that a polynomial has turning points. Recall that the degree of a Polynomial with one variable is the largest exponent of that variable. Example: a polynomial of Degree 4 will have 3 turning points or less x 4 2x 2 +x has 3 turning points x 4 2x has only 1 turning point The most is 3, but there can be less. We can look at a graph and count the number of turning points and simply add 1 and that will be the least degree of the polynomial.
60 Identify the zeros (x-intercepts), maximums and minimums of 3 2 f ( x) x 2x 5x 1 and f x x x x 4 3 2 ( ) 2 5 4 6 A rectangular piece of sheet metal is 10 in. long and 10 in. wide. Squares of side length x are cut from the corners and the remaining piece is folded to make an open top box. a) What size square can be cut from the corners to give a box with a volume of 25 cubic inches. b) What size square should be cut to maximize the volume of the box? What is the largest possible volume of the box?
An open box is to be made from a rectangular piece of cardboard that is 12 by 6 feet by cutting out squares of side length x ft from each corner and folding up the sides. a) Express the volume of the box vxas ( ) a function of the size x cut out at each corner. 61 b) Use your calculator to determine what size square can be cut from the corners to give a box with a volume of 40 cubic feet. c) Use your calculator to approximate the value of x which will maximize the volume of the box. Use your Graphing Calculator to find the appropriate polynomial model that fits the data. Use it to make predictions. x 1 2 3 4 5 6.. 10 f(x) 26-4 -2 2 2 16? The table shows the average price (in thousands of dollars) of a house in the Northeastern United States for 1987 to 1995. Find a polynomial model for the data. Then predict the average price of a house in the Northeast in 2000. x 1987 1988 1989 1990 1991 1992 1993 1994 1995 f(x) 140 149 159.6 159 155.9 169 162.9 169 180
62 Polynomial Inequalities with degree two or more. 1. Set one side of the inequality equal to zero. 2. Temporarily convert the inequality to an equation. 3. Solve the equation for x. If the equation is a rational inequality, also determine the values of x where the expression is undefined (where the denominator equals zero). These are the partition values. 4. Plot these points on a number line, dividing the number line into intervals. 5. Choose a convenient test point in each interval. Only one test point per interval is needed. 6. Evaluate the polynomial at these test points and note whether they are positive or negative. 7. If the inequality in step 1 reads 0, select the intervals where the test points are positive. If the inequality in step 1 reads 0, select the intervals where the test points are negative. Quadratic Inequalities Example 1. Solve. 2 a) x 2x 15 0 b) x 2 2x 15 0 c) 2 3x 11x 1 5 More Polynomial Inequalities Example 2. Solve. a) ( x 5)(3x 4)( x 2) 0 b) 3 16x x 0 c) 2 ( x 5) ( x 1) 0 d) 3 2 x x x 3 16 48 e) 2 2 x x (2 3) 0