# Lesson 7.1 Polynomial Degree and Finite Differences

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1 Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b c Determine which of the epressions are polnomials. For each polnomial, state its degree and write it in general form. If it is not a polnomial, eplain wh not. a b. 1 c. 3. The figures below show wh the numbers in the sequence 1, 3, 6, 10,... are called triangular numbers. a. Complete the table. n nth triangular number b. Calculate the finite differences for the completed table. c. What is the degree of the polnomial function that ou would use to model this data set? d. Write a polnomial function t that gives the nth triangular number as a function of n. (Hint: Create and solve a sstem of equations to find the coefficients.) Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

3 Lesson 7.3 Completing the Square 1. Factor each quadratic epression. a. 10 b. 1 4 c What value is required to complete the square? a. 18 b. c Convert each quadratic function to verte form b completing the square. a b c. 4. Find the verte of the graph of each quadratic function, and state whether the verte is a maimum or a minimum. a. ( )( 6) b c Rewrite each epression in the form a b c, and then identif the coefficients a, b, and c. a b. ( 8) c. ( 3)( ) 6. A ball is thrown up and off the roof of a 7 m tall building with an initial velocit of 14.7 m/s. a. Let t represent the time in seconds and h represent the height of the ball in meters. Write an equation that models the height of the ball. b. At what time does the ball reach maimum height? What is the ball s maimum height? c. At what time does the ball hit the ground? Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

4 Lesson 7.4 The Quadratic Formula 1. Evaluate each epression. Round our answers to the nearest thousandth. a (1)( ) b. 4 ( 4) 4()(1) (1) () c. ( ) 4(4)( 3) (4) d ()() (). Solve b an method. Give our answers in eact form. a b. 1 c d e..8 f Use the quadratic formula to find the zeros of each function. Then, write each equation in factored form, a r 1 r, where r 1 and r are the zeros of the function. a. 4 b. 8 6 c Write a quadratic function in general form that satisfies the given conditions. a. a 1; -intercepts of graph are 4 and b. -intercepts of graph are 0 and 13; graph contains point (, ) c. -intercept of graph is 4.8; -intercept is CHAPTER 7 Discovering Advanced Algebra More Practice Your Skills 010 Ke Curriculum Press

5 Lesson 7. Comple Numbers 1. Add, subtract, or multipl. a. ( 6i ) (1 i ) b. (.4.6i) (.9 1.8i) c. 4i( 6 i ) d. (. 1.i)( i). Find the conjugate of each comple number. a. 4i b. 7i c i 3. Rewrite each quotient in the form a bi. a. b. 1 i 3 i 1 i c. 3 i 6i 4. Solve each equation. Use substitution to check our solutions. Label each solution as real, imaginar, and/or comple. a. 0 b. 7 0 c. ( ) 1 d. 1 0 e f. ( 7)( 3). Write a quadratic function in general form that has the given zeros and leading coefficient of 1. a. 4, 7 b. 11i, 11i c. 3i, 3i 6. Name the comple number associated with each point, A through C, on the comple plane shown. C Imaginar A Real B Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

6 Lesson 7.6 Factoring Polnomials 1. Without graphing, find the -intercepts and the -intercept for the graph of each equation. a. ( 8) b. 3( 4)( ) c. 0.7( )( 6). Write the factored form of the quadratic function for each graph. Don t forget the vertical scale factor. a. b. (0, 4) ( 3, 0) (4, 0) ( 4, 0) (, 0) 0 (0, 4) 3. Convert each polnomial function to general form. a. (.)(.) b. 0.( 3) c. ( 1)( 1) 4. Write each polnomial as a product of factors. Some factors ma include irrational numbers. a b. 3 3 c. 169 d. 1 e f Sketch a graph for each situation if possible. a. A quadratic function with two real zeros, whose graph has the line as its ais of smmetr b. A cubic function with three real zeros, whose graph has a positive -intercept 46 CHAPTER 7 Discovering Advanced Algebra More Practice Your Skills 010 Ke Curriculum Press

7 Lesson 7.7 Higher-Degree Polnomials 1. Refer to these two graphs of polnomial functions. i. ii ( 3, 0) (, 0) (0, 0) (0, 4) (, 0) (, 0) 10 0 a. Identif the zeros of each function. b. Find the -intercept of each graph. c. Identif the lowest possible degree of each polnomial function. d. Write the factored form for each polnomial function. Check our work b graphing on our calculator.. Write a polnomial function with the given features. a. A quadratic function whose graph has verte (3, 8), which is a minimum, and two -intercepts, one of which is b. A fourth-degree polnomial function with two double roots, 0 and, and whose graph contains the point (1, 1) 3. Write the lowest-degree polnomial function that has the given set of zeros and whose graph has the given -intercept. Write each polnomial function in factored form. Give the degree of each function. a. Zeros: 3, ; -intercept: 30 b. Zeros: i, (double root), ; -intercept: 80 Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

8 Lesson 7.8 More About Finding Solutions 1. Divide. a. ) b. 4 ) Varsha started out dividing two polnomials b snthetic division this wa: a. Identif the dividend and divisor. b. Write the numbers that will appear in the second line of the snthetic division. c. Write the numbers that will appear in the last line of the snthetic division. d. Write the quotient and remainder for this division. 3. In each division problem, use the polnomial that defines P as the dividend and the binomial that defines D as the divisor. Write the result of the division in the form P() D() Q() R, where the polnomial that defines Q is the quotient and R is an integer remainder. (It is not necessar to write the remainder if R 0.) a. P() 9 ; D() b. P() 3 8 ; D() 1 4. Make a list of the possible rational roots of each equation. a b Find all the zeros of each polnomial function. Then write the function in factored form. a b CHAPTER 7 Discovering Advanced Algebra More Practice Your Skills 010 Ke Curriculum Press

9 d. Vertices: 14 3, 0, (6, 0), (0, 4.), (0, 3.) e. 14 3, 0 : \$140; (6, 0): \$180; (0, 4.): \$147; (0, 3.): \$1.0 f. 6 batches of full sheet cakes, 0 batches of halfsheet cakes; \$180 LESSON 7.1 Polnomial Degree and Finite Differences 1. a. b. 3 c.. a. Polnomial; 4; a. b. Not a polnomial; 1 has a negative eponent c. Polnomial; 0; alread in general form n nth triangular number b. D 1 {, 3, 4,, 6, 7}, D {1, 1, 1, 1, 1} c. d. t (n) 1 n 1 n, or t (n) 0.n 0.n LESSON 7. Equivalent Quadratic Forms 1. a. General form b. Verte form c. Factored form. a. 6 9 b. c a. (0, 0) b. (1, 6) c. ( 4, 6.) 4. a. 1 and 6 b. 0 and c. 7.. a. 1. b. ( 1., ); minimum c. ( 1.) LESSON 7.3 Completing the Square 1. a. ( ) b. 1 c. (3 4). a. 81 b., or 6. c a. ( 7) 1 b. ( 1) 8 c. ( 1.) a. (, 16); minimum b. ( 1, 3.); maimum c. ( 4., 30.); minimum. a. 3 6 ; a 3, b 6, c b. 16; a, b 16, c 0 c. 7 1; a, b 7, c 1 6. a. h 4.9t 14.7t 7 b. 1. s; 86.0 m c..69 s LESSON 7.4 The Quadratic Formula 1. a b c d a. or b. 3 or 4 c. 7 d e. 0 or.8 f a. ( 8)( 3) b. ( 3)( 1) c. 4( 1)( 0.) 4. a. 6 8 b. 13 c LESSON 7. Comple Numbers 1. a. 6 7i b i c. 4 4i d i. a. 4i b. 7i c i 3. a. 3 1 i, or i b. i, or 0 i c. 6 1 i 4. a. 1 i; comple b. i 7 ; imaginar and comple c. 9 ; real and comple 1 i d. 3 ; comple e. 3_ i, or 1.i; imaginar and comple 6 f. 140, or 3 3 ; real and comple. a b c A: 3i, or 0 3i ; B: 3i ; C: 4 i Discovering Advanced Algebra More Practice Your Skills ANSWERS Ke Curriculum Press

10 LESSON 7.6 Factoring Polnomials 1. a. -intercept: 8; -intercept: 64 b. -intercepts: 4, ; -intercept: 4 c. -intercepts: 0,, 6; -intercept: 0. a. ( 3)( 4) b. 0.( 4)( ) 3. a. 1. b c a. ( 7) b. ( 1)( ) c. ( 13i)( 13i) d. 1 1 e. ( 1)( 1)( 3)( 3) f. 3( 1)( 4)( ). Possible graphs: a. b. 3. a. 9 ( )( 1) 7 b. 3 8 ( 1) 3 4. a. 1,, 4, 8 b. 1,, 3,, 6, 10, 10, 1, 30, 1 3,,, 1. a., 3i, and 3i; ( )( 3i)( 3i), or ( ) 9 b., 4_ 3, and 1_ ; 6( ) 4_ 3 1_, or ( )(3 4)( 1) LESSON 8.1 Using the Distance Formula 1. a. 13 units b. units c. 0 units, or units d. 80 units, or 4 units e. a 9 units f. units. a. or 14 b. 4 7 LESSON 7.7 Higher-Degree Polnomials 1. a. i. 3, 0, ; ii., b. i. 0; ii. 4 c. i. 3; ii. 4 d. i. ( 3)( ); ii. 0.( ) ( ). a. ( 1)( ), or 1 10 b. ( ), or a. ( 3)( ); degree b. ( i )( i )( ) ( ), or 4 ( ) ( ); degree a. ( ) ( 3) LESSON 7.8 More About Finding Solutions 1. a. 3 1 b a. Dividend: ; divisor: 3 b c d. Quotient: ; remainder: 4 b ANSWERS Discovering Advanced Algebra More Practice Your Skills 010 Ke Curriculum Press

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