Using Properties of Exponents
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1 6.1 Using Properties of Exponents Goals p Use properties of exponents to evaluate and simplify expressions involving powers. p Use exponents and scientific notation to solve real-life problems. VOCABULARY Scientific notation PROPERTIES OF EXPONENTS Let a and b be real numbers and let m and n be integers. Product of powers property a m p a n a Power of a power property (a m ) n a Power of a product property (ab) m a b Negative exponent property a m, a 0 Zero exponent property a 0, a 0 Quotient of powers property a m an a, a 0 Power of a quotient property a b m, b 0 Example 1 Evaluating Numerical Expressions 6 a b. 3 2 c. (3 2 ) Algebra 2 Notetaking Guide Chapter 6
2 Example 2 3 a. x 4 y2 Simplifying Algebraic Expressions Power of a quotient property Power of a power property Negative exponent property b. (3a) 3 a 9 a 7 a 9 a 7 Power of a product property Product of powers property Simplify exponent. c. (c c d) d2 c 9 d 2 c 9 d 2 Power of a product property Power of a power property Quotient of powers property Simplify exponents. Zero exponent property Negative exponent property Checkpoint Complete the following exercises. 1. Evaluate (2 2 ) 3 (2 5 jk ) ). 2. Simplify (j ( 1. 2 k) 3 2 Lesson 6.1 Algebra 2 Notetaking Guide 125
3 Example 3 Comparing Real-Life Volumes The radius of a basketball is about 5.7 times greater than the radius of a golf ball. How many times as great as the golf ball s volume is the basketball s volume? Let r represent the radius of the golf ball. Basketball s volume Golf ball s volume 4 3 π πr3 The volume of a sphere is 4 3 πr π 4 3 πr3 Power of a product property The basketball s volume is about golf ball s volume. Quotient of powers property Zero exponent property Approximate power. times as great as the Example 4 Using Scientific Notation in Real Life Greenland covers about square kilometers and has approximately people. About how many square kilometers are there per person? Homework Land area P opulation There are about Divide land area by population. Quotient of powers property Use a calculator. Write in standard notation. square kilometers per person. 126 Algebra 2 Notetaking Guide Chapter 6
4 6.2 Evaluating and Graphing Polynomial Functions Goals p Evaluate polynomial functions. p Graph polynomial functions. Your Notes VOCABULARY Polynomial function Leading coefficient Constant term Degree Standard form of a polynomial function End behavior Example 1 Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type, and leading coefficient. a. f(x) 2x 1.2 2x 3 4x 2 b. f(x) 0.32x x 3 71 a. The function a polynomial function because does not have a. b. The function a polynomial function. Its standard form is. It has degree, so it is a function. The leading coefficient is. Lesson 6.2 Algebra 2 Notetaking Guide 127
5 Checkpoint Complete the following exercise. 1. State the degree, type, and leading coefficient of f(x) x 4 2x 3 4x 2 x 5. Example 2 Using Synthetic Substitution Use synthetic substitution to evaluate f(x) 2x 4 4x 3 x 2 1 when x 2. Write the value of x and the coefficients of f(x) as shown. Bring down the leading coefficient. Multiply by and write the result in the next column. the numbers in that column and write the sum below the line. Continue to multiply and add. x-value 2x 4 (4x 3 ) x 2 0x 1 Polynomial in standard form Coefficients f(2) END BEHAVIOR FOR POLYNOMIAL FUNCTIONS The graph of f(x) a n x n a n 1 x n 1... a 1 x a 0 has this end behavior: For a n > 0 and n even, f(x) as x and f(x) as x. For a n > 0 and n odd, f(x) f(x) as x. For a n < 0 and n even, f(x) f(x) as x. For a n < 0 and n odd, f(x) f(x) as x. as x and as x and as x and 128 Algebra 2 Notetaking Guide Chapter 6
6 Example 3 Graphing Polynomial Functions Graph (a) f(x) x 3 2x 2 x 1 and (b) f(x) x 4 4x 2 x 1. a. Make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. x f (x) x 1 The degree is and the leading coefficient is, so f(x) as x and f(x) as x. b. Make a table of values and y plot the corresponding points. 3 Connect the points with a smooth curve and check the end behavior y x x 1 f (x) The degree is and the 5 leading coefficient is, so f(x) as x and f(x) as x. 3 Checkpoint Complete the following exercises using the function f(x) x 4 x 3 4x 2 4x 3. Homework 2. Evaluate f(x) for x 4 3. Graph f(x). using synthetic substitution. y x 2 6 Lesson 6.2 Algebra 2 Notetaking Guide 129
7 6.3 Adding, Subtracting, and Multiplying Polynomials Goals p Add, subtract, and multiply polynomials. p Use polynomial operations in real-life problems. Your Notes Example 1 Adding Polynomials Vertically and Horizontally a. 2x 3 3x 2 7x 5 3x 3 2x 2 3x b. (x 3 6x 4) (6x 3 2x 2 9x 2) Example 2 Subtracting Polynomials Vertically and Horizontally a. 7x 3 4x 2 x 4 7x 3 4x 2 x 4 (5x 3 x 2 x 6) b. (3x 4 9x 3 2) (2x 4 6x 3 2x 4) 3x 4 9x 3 2 Example 3 Multiplying Polynomials Vertically x 2 x 3 x 2 Multiply x 2 x 3 by 2. Multiply x 2 x 3 by x. Combine like terms. 130 Algebra 2 Notetaking Guide Chapter 6
8 Example 4 Multiplying Polynomials Horizontally Multiply the polynomials. (x 2)(3x 2 x 5) (x 2) (x 2) (x 2) Example 5 Multiplying Three Binomials Multiply the polynomials. (x 2)(x 1)(x 3) ( )(x 3) ( )(x) ( )(3) SPECIAL PRODUCT PATTERNS Sum and Difference Example (a b)(a b) a 2 b 2 (x 3)(x 3) Square of a Binomial (a b) 2 a 2 2ab b 2 (y 4) 2 (a b) 2 a 2 2ab b 2 (3t 2 2) 2 Cube of a Binomial (a b) 3 (x 1) 3 a 3 3a 2 b 3ab 2 b 3 (a b) 3 (p 2) 3 a 3 3a 2 b 3ab 2 b 3 Lesson 6.3 Algebra 2 Notetaking Guide 131
9 Example 6 Using Special Product Patterns Multiply the polynomials. a. (3z 4)(3z 4) ( ) 2 2 b. (4x 2 3y) 2 ( ) 2 2( )( ) ( ) 2 c. (cd 4) 3 ( ) 3 3( ) 2 ( ) 3( )( ) 2 3 Checkpoint Complete the following exercises. 1. Add (3x 2 3x 7) (x 2 9x 5). 2. Subtract (6x 3 x 2 1) (2x 3 6x 3). 3. Multiply (x 3)(x 1)(x 7). Homework 4. Multiply (2w 3) Algebra 2 Notetaking Guide Chapter 6
10 6.4 Factoring and Solving Polynomial Equations Goals p Factor polynomial expressions. p Use factoring to solve polynomial equations. Your Notes VOCABULARY Factor by grouping Quadratic form SPECIAL FACTORING PATTERNS Sum of Two Cubes a 3 b 3 (a b)(a 2 ab b 2 ) Example x 3 8 (x 2)( ) Difference of Two Cubes a 3 b 3 (a b)(a 2 ab b 2 ) Example 8x 3 1 (2x 1)( ) Example 1 Factoring the Sum or Difference of Cubes Factor each polynomial. a. x 3 64 x 3 3 (x )( ) b. 54y 4 16y 2y( ) 2y ( )3 3 2y( )( ) Lesson 6.4 Algebra 2 Notetaking Guide 133
11 Example 2 Factoring by Grouping Factor the polynomial x 3 3x 2 4x 12. x 3 3x 2 4x 12 x 2 ( ) 4( ) Factor by grouping. Difference of squares Example 3 Factoring Polynomials in Quadratic Form Factor (a) 16x 4 1 and (b) 2x 6 10x 4 12x 2. a. 16x 4 1 ( ) 2 2 b. 2x 6 10x 4 12x 2 2x 2 ( ) Example 4 Solving a Polynomial Equation Solve x 4 4 5x 2. x 4 4 5x Write original equation. Rewrite in standard form. Factor trinomial. 0 Factor difference of squares. x, x, x, or x Zero product property The solutions are. Check these in the original equation. 134 Algebra 2 Notetaking Guide Chapter 6
12 Checkpoint Factor each polynomial in Exercises x x 3 x 2 2x 2 3. x 4 7x Solve x 5 2x x 3. Example 5 Solving a Polynomial Equation in Real Life A rectangular swimming pool has a volume of 512 cubic feet. The pool s dimensions are x feet deep by 6x 8 feet long by 6x 16 feet wide. How deep is the pool? Verbal Model Volume p Length p Width Labels Volume (cubic feet) Depth (feet) Length (feet) Width (feet) Homework Algebraic Model Standard form 0 Factor by grouping. 0 The only real solution is x, so 6x 8 and 6x 16. The pool is feet deep. The dimensions are. Lesson 6.4 Algebra 2 Notetaking Guide 135
13 6.5 The Remainder and Factor Theorems Goals p Divide polynomials and relate the result to the remainder theorem and the factor theorem. p Use polynomial division in real-life problems. Your Notes VOCABULARY Polynomial long division Synthetic division Example 1 Using Polynomial Long Division Divide 4x 4 x 2 18x 8 by x 2 2x 3. Write division in the same format you would use when dividing numbers. Include a 0 as the coefficient of x 3. x 2 2x 34x 4 0x 3 12x 2 18x 8 Write the result as follows. 4x 4 x 2 18x 8 x 2 2x Algebra 2 Notetaking Guide Chapter 6
14 REMAINDER THEOREM If a polynomial f(x) is divided by x k, the remainder is r. Example 2 Using Synthetic Division Divide x 3 x 2 5x 3 by x 2. To find the value of k, rewrite the divisor in the form x k. Because x 2 x, k x 3 x 2 5x 3 x 2 FACTOR THEOREM A polynomial f(x) has a factor x k if and only if f(k). Example 3 Factoring a Polynomial Factor f(x) x 3 19x 30 given that f(5) 0. Because f(5) 0, you know that is a factor of f(x). Use synthetic division to find the other factors The result gives the coefficients of the quotient. x 3 19x 30 ( )( ) ( )( )( ) Lesson 6.5 Algebra 2 Notetaking Guide 137
15 Example 4 Finding Zeros of a Polynomial Function A zero of f(x) x 3 x 2 4x 4 is x 1. Find the other zeros. Because f(1) 0, you know that is a factor of f(x). Use synthetic division to find the other factors The result gives the coefficients of the quotient. f(x) x 3 x 2 4x 4 ( )( ) ( )( )( ) By the factor theorem, the zeros of f are. Checkpoint Complete the following exercises. 1. Use long division to divide x 2 4x 1 by x Use synthetic division to divide 2x 3 x 2 3x 4 by x 1. Homework 3. Factor f(x) 2x 3 x 2 25x 12 given that f(4) A zero of f(x) x 4 5x 2 4 is 1. Find the other zeros. 138 Algebra 2 Notetaking Guide Chapter 6
16 6.6 Finding Rational Zeros Goals p Find the rational zeros of a polynomial function. p Use polynomial equations to solve real-life problems. THE RATIONAL ZERO THEOREM If f(x) a n x n... a 1 x a 0 has coefficients, then every rational zero of f has the following form: p q factor of constant term factor of leading coefficient Example 1 Using the Rational Zero Theorem Find the rational zeros of f(x) x 3 5x 2 2x 8. List the possible rational zeros. The leading coefficient is and the constant term is. So, the possible rational zeros are: x,,, Test these zeros using synthetic division. Test x is a zero. Because is a zero of f, write f(x). Factor the trinomial and use the factor theorem. f(x) The zeros of f are. Lesson 6.6 Algebra 2 Notetaking Guide 139
17 Example 2 Using the Rational Zero Theorem Find all real zeros of f(x) 6x 4 7x 3 19x 2 5x 6. List the possible rational zeros of f:. Choose values to check by using your graphing utility to graph the function. Two reasonable choices are x and x. Check the value using synthetic division is a zero. Factor out a binomial using the result of the synthetic division. f(x) Rewrite as product of factors. Factor from the second factor. Multiply the first factor by. Repeat the steps above for g(x). Any zero of g will also be a zero of f. The possible rational zeros of g are x. Confirm that the value x is a zero by using synthetic division. f(x) Find the remaining zeros of f by using the quadratic formula to solve. The real zeros of f are,,, and. 140 Algebra 2 Notetaking Guide Chapter 6
18 Checkpoint Complete the following exercise. 1. Find all real zeros of f(x) 5x 4 6x 3 24x 2 15x 2. Example 3 Writing and Using a Polynomial Model You are making a wooden rectangular box. You want the volume of the box to be 135 cubic inches. You want the length of each side of the square base to be x inches and the height to be x 12 inches. What are the dimensions? The volume is V Bh where B base area and h height. Verbal Model Labels Volume Volume Area of Base Height p (cubic inches) (square inches) (inches) Height Algebraic Model The possible rational solutions are x,,,,,,, and. Homework In this case, it makes sense to test only positive x-values. 1 3 So, x is a solution. The base should be inches by inches. The height should be inches. Lesson 6.6 Algebra 2 Notetaking Guide 141
19 6.7 Using the Fundamental Theorem of Algebra Goals p Use the fundamental theorem of algebra. p Use technology to approximate zeros. Your Notes VOCABULARY Repeated solution For the equation f(x) 0, k is a repeated solution if and only if the factor (x k) has degree greater than 1 when f is factored completely. Example 1 Finding the Number of s or Zeros a. The equation x 3 2x 2 x 2 0 has three solutions: 2, i, i. b. The function f(x) x 4 3x 3 4x 2 has four zeros: 0, 0, 1, 4. Example 2 Finding the Zeros of a Polynomial Function Find all the zeros of f(x) x 5 2x 4 3x 3 6x 2 4x 8. The possible rational zeros are 1, 2, 4, and 8. Using synthetic division, you can determine that 2 is a repeated zero and that 2 is also a zero. You can write the function in factored form as follows: f(x) (x 2)(x 2)(x 2)(x 2 1). Complete the factorization. f(x) (x 2)(x 2)(x 2)(x i)(x i) The five zeros are 2, 2, 2, i, and i. The graph of f is shown at the right. Note that only the real zeros appear as x-intercepts. Also note that the graph only touches the x-axis at the repeated zero x 2, but crosses the x-axis at the zero x x 5 y 142 Algebra 2 Notetaking Guide Chapter 6
20 Checkpoint Complete the following exercises. 1. State the number of zeros of f(x) x 3 3x 2 5x 25 and tell what they are. 2. Find all zeros of f(x) x 4 7x 2 18x 10. Example 3 Using Zeros to Write Polynomial Functions Write a polynomial function f of least degree that has real coefficients, leading coefficient 1, and zeros 1 and 2 i. Because the coefficients are real and 2 i is a zero, must also be a zero. Use the three zeros and the factor theorem to write f(x) as a product of three factors. f(x) ( )[x ( )][x ( )] Factored form ( )[ ][ ] Regroup. Multiply. Expand, use i 2. Simplify. Multiply. Combine like terms. Check You can check by evaluating f(x) at each of its zeros. Lesson 6.7 Algebra 2 Notetaking Guide 143
21 Checkpoint Complete the following exercise. 3. Write a polynomial function of least degree that has real coefficients, a leading coefficient of 1, and 4, 3i, and 3i as zeros. Example 4 Approximating Real Zeros Approximate the real zeros of f(x) x 4 5x 3 6x 2 20x 8. Use a graphing calculator to approximate the real zeros of the function. Use the Zero (or Root) feature. You can see that the real zeros are about. The polynomial function has degree, so there must be other zeros. These may be repeats of the real zeros or imaginary. In this case, they are : x. Homework Checkpoint Complete the following exercise. 4. Approximate the real zeros of f(x) x 5 6x 4 10x 3 18x 2 21x. 144 Algebra 2 Notetaking Guide Chapter 6
22 6.8 Analyzing Graphs of Polynomial Functions Goals p Analyze the graph of a polynomial function. p Use polynomial functions in real life. Your Notes VOCABULARY Local maximum Local minimum ZEROS, FACTORS, SOLUTIONS, AND INTERCEPTS Let f(x) a n x n a n 1 x n 1... a 1 x a 0 be a polynomial function. The following statements are equivalent. Zero: is a zero of the polynomial function f. Factor: is a factor of the polynomial f(x). : is a solution of the polynomial equation f(x) 0. If k is a real number, then the following is also equivalent. x-intercept: is an x-intercept of the graph of the polynomial function f. Lesson 6.8 Algebra 2 Notetaking Guide 145
23 Example 1 Using x-intercepts to Graph a Polynomial Function Graph the function f(x) 1 2 (x 1)2 (x 3). Plot x-intercepts. Because x 1 and x 3 are factors of f(x), and are the x-intercepts of the graph of f. Plot the points (, ) and (, ). Plot points between and beyond the x-intercepts. x y Determine the end behavior of the graph. Because f(x) has linear factors of the form x k and a constant factor of, it is a function with a leading coefficient. Therefore, f(x) f(x) as x. Draw the graph so that it passes through the points you plotted and has the appropriate end behavior. as x and y x 1 TURNING POINTS OF POLYNOMIAL FUNCTIONS The graph of every polynomial function of degree n has at most turning points. Moreover, if a polynomial function has n distinct real zeros, then its graph has exactly turning points. 146 Algebra 2 Notetaking Guide Chapter 6
24 Example 2 Finding Turning Points Graph each function. Identify the x-intercepts and the points where the local maximums and local minimums occur. a. f(x) x 3 4x 2 x 4 b. f(x) x 4 2x 3 2x 2 3x a. Use a graphing calculator to graph the function. Notice that the graph has x-intercepts and turning points. You can use the graphing calculator s Zero, Maximum, and Minimum features to approximate the coordinates of the points. The x-intercepts of the graph are. The function has a local minimum at (, ) and a local maximum at (, ). b. Use a graphing calculator to graph the function. Notice that the graph has x-intercepts and turning points. You can use the graphing calculator s Zero, Maximum, and Minimum features to approximate the coordinates of the points. The x-intercepts of the graph are. The function has local maximums at (, ) and (, ), and it has a local minimum at (, ). Lesson 6.8 Algebra 2 Notetaking Guide 147
25 Checkpoint Complete the following exercises. 1. Graph f(x) (x 1)(3x 1)(2x 5). 6 y x 6 2. Use a graphing calculator to identify the x-intercepts, local maximums, and local minimums of the graph of f(x) x 4 x 3 6x 2 4x 2. Homework 148 Algebra 2 Notetaking Guide Chapter 6
26 6.9 Modeling with Polynomial Functions Goals p Use finite differences. p Use technology to find polynomial models. Your Notes VOCABULARY Finite differences Example 1 Writing a Cubic Function Write the cubic function whose graph is shown at the right. Use the three given x-intercepts to write f(x) a. To find a, substitute the coordinates of the fourth point. (1, 0) 1 (2, 0) a, so a. f(x) x (0, 4) Check Check the graph s end behavior. The degree of f is and a 0, so f(x) as x and f(x) as x. 5 y (2, 0) Lesson 6.9 Algebra 2 Notetaking Guide 149
27 Checkpoint Complete the following exercise. 1. Write the cubic function of the graph shown. (0, 4) y 1 (2, 0) 3 (1, 0) (1, 0) x Example 2 Finding Finite Differences An equation for a polynomial function is f(n) n 3 2n 2 3n 1. Show that this function has constant third-order differences. Write the first several function values. Find the first-order differences by subtracting consecutive function values. Then find the second-order differences by subtracting consecutive differences. Finally, find the third-order differences by subtracting consecutive differences. f(1) f(2) f(3) f(4) f(5) f(6) Function values for equally-spaced n-values First-order differences Second-order differences Third-order differences PROPERTIES OF FINITE DIFFERENCES 1. If a polynomial function f(x) has degree n, then the nth-order differences of function values for equally spaced x-values are. 2. Conversely, if the nth-order differences of equally-spaced data are, then the data can be represented by a polynomial function of degree n. 150 Algebra 2 Notetaking Guide Chapter 6
28 Example 3 Modeling with Finite Differences The values of a polynomial function for five consecutive whole numbers are given below. Write a polynomial function for f(n). f(1) 5, f(2) 14, f(3) 27, f(4) 44, and f(5) 65 Begin by finding the finite differences. f(1) f(2) f(3) f(4) f(5) Function values for equally-spaced n-values First-order differences Second-order differences Because the differences are constant, you know that the numbers can be represented by a function which has the form f(n). By substituting the first three values into the function, you can obtain a system of three linear equations in variables. Using a calculator to solve the system gives a, b, and c. The polynomial function is f(n). Checkpoint Complete the following exercise. Homework 2. Values of a polynomial function for six consecutive whole numbers are given. Write a polynomial function for f(n). f(1) 2, f(2) 8, f(3) 22, f(4) 50, f(5) 98, and f(6) 172 Lesson 6.9 Algebra 2 Notetaking Guide 151
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