Algebra 2 - Task 4.9
Name___________________________________
Unit 4 Review - Polynomials
Date________________ Period____
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1) Explain the difference between the graphs of functions with even degrees versus odd degrees. 3) How do you determine the degree of a polynomial? What is the connection between the degree and the roots of a polynomial? 5) Be able to classify a polynomial (state the first and last name).
2) How do you know when a polynomial is written in standard form? 4) What is the leading coefficient of a polynomial?
6) Explain your approach for factoring polynomials (What are the steps you take? How do you know when your polynomial is factored completely?) 7) What are all the factoring methods that you 8) When trying to find the roots of functions, why know? How do you know which one to use? do we always make f ( x) = 0? Answer using complete sentences. 9) When dividing polynomials, be able to explain 10) What are the main differences between long and justify how you know when a binomial is a division and synthetic division of polynomials? factor of a polynomial. Be able to explain the Why don't we always use synthetic division to connection between the remainder and a factor. divide polynomials? When do we know when to use each method? Write in Standard Form. Then name each polynomial by degree and number of terms. 11) 7k 2 − 8k 5 − 3 − 2k 13) 7k 3 − 3k − 2k 2 15) −10 p 17) −5 19) k − 9k 4 − 9k 5 − 10 + 5k 6 A. Describe the end behavior of each function.
12) 14) 16) 18) 20)
−2 − 8v − 5v 3 r4 1 4 − 4x 9 p4 − 8 − p2
21) f ( x) = x 4 − x 2 + x + 1 22) f ( x) = − x 3 + 3 x 2 − 4 23) f ( x) = − x 3 − 5 x 2 − 3 x + 4 24) f ( x) = x 2 + 4 x − 2 25) f ( x) = x 3 − 3 x 2 − 2 26) f ( x) = 2 x 2 + 8 x + 6 27) f ( x) = − x 5 + 4 x 3 − 4 x + 4 28) f ( x) = x 4 + x 3 − 2 x 2 + 1 B. Sketch the polynomial given its roots. (For the odd-numbered problems use "a"=negative, for the even-numbered problems use "a"=positive) 29) 3, −4, −1 30) −5, 2, −2 31) 5, −5, 0 32) 3, −4, 1 33) 4, −1, 0 34) 2, −1, 1 C. Construct the polynomial given the roots, then classify the polynomial (state the first and last name) 35) −3, −4, −2 37) −5, 5, 3 39) 4, −5, 2 Factor each polynomial completely.
36) −3, −4, 2 38) −4, 1, 0, −3, −5 40) 5, 1, −2
41) x 3 − 8 = 0
42) x 3 + 125 = 0
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Worksheet by Kuta Software LLC
43) x 3 + 1 = 0 45) x 3 + 27 = 0 47) x 3 + 8 = 0 49) x 3 − 1 = 0 51) 9n 2 − 16 53) 9b 2 − 1 55) 12 x 3 + 16 x 2 − 9 x − 12 57) 56 x 3 − 49 x 2 − 40 x + 35 Find all the roots.
44) 46) 48) 50) 52) 54) 56) 58)
x 3 − 64 = 0 x 3 − 27 = 0 x 3 + 64 = 0 x 3 − 125 = 0 9v 2 − 25 16 x 2 − 1 k 3 + 6k 2 − 4k − 24 12a 3 + 36a 2 + 10a + 30
59) f ( x) = 3 x 3 − x 2 + 6 x − 2 61) f ( x) = 3 x 3 − 6 x 2 + 4 x − 8 63) f ( x) = 3 x 3 − 6 x 2 + 5 x − 10 65) f ( x) = 8 x 3 + 125 67) f ( x) = 3 x 3 + 2 x 2 − 6 x − 4 Perform the indicated operation.
60) 62) 64) 66) 68)
f ( x) = 27 x 3 − 64 f ( x) = x 3 − 27 f ( x) = 8 x 3 − 27 f ( x) = 3 x 3 − 5 x 2 − 9 x + 15 f ( x) = 2 x 3 + 6 x 2 + 5 x + 15
69) g( x) = 3 x + 1 h( x) = x 2 − 3 Find ( g ⋅ h)( x) 71) g(a) = −2a − 3 h(a) = a 2 + 4 Find ( g − h)(a) 73) f (n) = 4n g(n) = n 2 + 4 Find ( f + g)(n) 75) f ( x) = x 2 + 5 g( x) = 4 x − 2 Find ( f + g)( x) 77) g( x) = 3 x h( x) = 2 x 2 − 3 x Find ( g + h)( x) a) Divide using long division factor
70) g( x) = x 3 + 2 x h( x) = 3 x − 3 Find ( g − h)( x) 72) f (a) = −2a 3 + 5 g(a) = a − 4 Find ( f − g)(a) 74) g( x) = − x 2 + x h( x) = − x + 2 Find ( g − h)( x) 76) g( x) = − x + 5 f ( x) = x 2 − 2 Find ( g + f )( x) 78) g(a) = 3a + 4 h(a) = a 2 + 4 Find ( g + h)(a) b) Divide using synthetic division c) Determine if the binomial divisor is a
79) (a 4 − 3a 3 − 18a 2 − 80a + 71) ÷ (a − 7) 80) (b 3 − 3b 2 − 62b + 65) ÷ (b − 9) 81) ( x 4 − 3 x 3 − 73 x 2 + 27 x + 24) ÷ ( x − 10) 82) (9 x 4 − 46 x 3 + 49 x 2 − 34 x − 9) ÷ ( x − 4) 83) (n 3 − n 2 − n − 6) ÷ (n − 2) 84) (v 3 − v 2 − v − 16) ÷ (v − 3) Divide, and determine whether the binomial is a factor of the polynomial. 85) (56b 3 + 14b 2 + 7) ÷ (8b + 2) 87) (80 x 4 − 28 x 3 − 10 x 2 + 12 x − 55) ÷ (10 x + 9) 89) (4r 3 − 23r 2 − 23r + 10) ÷ (4r + 5) Previous Material - Complex Numbers.
86) (6n 3 + 24n 2 + 9n − 16) ÷ (3n + 3) 88) (16 x 3 + 8 x 2 + 7) ÷ (4 x + 2) 90) (9 p 3 + 62 p 2 − 34 p + 2) ÷ (9 p − 1)
91) (4 − 4i) + (2 + 8i)
92) (4 + 7i) − 5 − 5
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Worksheet by Kuta Software LLC
93) (1 − 4i) + (4 + 7i) 95) (−8i)(−8i)(6 − 8i) −9 + 4i 97) −2i −2 − 6i 99) 10i −9 − 4i 101) −3 − 10i 9 + 3i 103) −7 − 4i Graph each number in the complex plane.
94) (3 − 4i) − (6 + 4i) 96) (6i)(8i)(1 + 8i) −8 + 5i 98) −7i −3 + 2i 100) −5i −9 + i 102) −9 − 9i 1 + 4i 104) 1 − 3i
105) −3 − i
106) 1 − 5i Imaginary
Imaginary
Real
Real
107) 0
108) 5 − 2i Imaginary
Imaginary
Real
Real
Identify each complex number graphed. 109)
110)
Imaginary
Real
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Imaginary
Real
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