Algebra 2 5 Polynomials and Polynomial Functions Practice Problems Page 1 of 12 5.1 Use Properties of Exponents Evaluate the expression. Tell which properties of exponents you used. 1. 3 3 ⋅3 2 2. 5 2 5 5 3. 3 4 3 −2 Simplify the expression. Tell which properties of exponents you used. 4. (2 2 3 ) 5 5. ( 3 −2 )( 6 −1 ) 6. (3 3 5 ) −3 7. 3 3 9 −1 8. 2 3 −4 3 5 −2 9. 2 −3 3 2 ⋅ 2 −4 Describe and correct the error in simplifying the expression. 10. 10 2 = 5 11. (−3) 2 (−3) 4 =9 6 Write an expression that makes the statement true. 12. 15 12 8 = 4 7 11 ⋅ ? Write an expression for the figure’s area or volume in terms of x. 13. = 2 ℎ Mixed Review 14. (4.10) Write a quadratic function in vertex form whose graph has vertex (-4, 1) and passes through (-2, 5). 15. (4.8) Solve 2 − 2 − 35 = 0 using any algebraic method. 16. (4.6) Solve 3 2 − 7 = −31 using any algebraic method. 17. (3.6) Multiply [ 5 0 −4 1 ][ −3 2 6 2 ]. 18. (3.2) Solve { 4 −2 = −16 −3 +4 = 12 19. (2.4) Write an equation of the line that passes through the point (3, -1) and has the slope = −3. 20. (1.7) Solve | − 5| = 3. 5.2 Evaluate and Graph Polynomial Functions 1. Identify the degree, type, leading coefficient, and constant term of the polynomial function () = 6 + 2 2 − 5 4 . 2. Decide whether the function is a polynomial function. If so, write it is standard form and state its degree, type, and leading coefficient. () = 4 + √6 Use direct substitution to evaluate the polynomial function for the given value of x. 3. () = 5 3 − 2 2 + 10 − 15; = −1 4. ℎ() = + 1 2 4 − 3 4 3 + 10; = −4 Use synthetic substitution to evaluate the polynomial function for the given value of x. 5. () = 3 + 8 2 − 7 + 35; = −6 6. ℎ() = −7 3 + 11 2 + 4; = 3 Describe the degree and leading coefficient of the polynomial function whose graph is shown. 2
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Algebra 2 5 Polynomials and Polynomial Functions Practice Problems
Page 1 of 12
5.1 Use Properties of Exponents Evaluate the expression. Tell which properties of exponents you used.
1. 33 ⋅ 32
2. 52
55
3. 34
3−2
Simplify the expression. Tell which properties of exponents you used. 4. (22𝑦3)5
5. (𝑤3𝑥−2)(𝑤6𝑥−1)
6. (3𝑎3𝑏5)−3
7. 3𝑐3𝑑
9𝑐𝑑−1
8. 2𝑎3𝑏−4
3𝑎5𝑏−2
9. 𝑥2𝑦−3
3𝑦2 ⋅𝑦2
𝑥−4
Describe and correct the error in simplifying the expression.
10. 𝑥10
𝑥2 = 𝑥5 11. (−3)2(−3)4 = 96
Write an expression that makes the statement true. 12. 𝑥15𝑦12𝑧8 = 𝑥4𝑦7𝑧11 ⋅ ?
Write an expression for the figure’s area or volume in terms of x. 13. 𝑉 = 𝜋𝑟2ℎ
Mixed Review
14. (4.10) Write a quadratic function in vertex form whose graph has vertex (-4, 1) and passes through (-2, 5).
15. (4.8) Solve 𝑥2 − 2𝑥 − 35 = 0 using any algebraic method.
16. (4.6) Solve 3𝑥2 − 7 = −31 using any algebraic method.
17. (3.6) Multiply [5 0
−4 1] [
−3 26 2
].
18. (3.2) Solve {4𝑥 −2𝑦 = −16
−3𝑥 +4𝑦 = 12
19. (2.4) Write an equation of the line that passes through the point (3, -1) and has the slope 𝑚 = −3.
20. (1.7) Solve |𝑓 − 5| = 3.
5.2 Evaluate and Graph Polynomial Functions 1. Identify the degree, type, leading coefficient, and constant term of the polynomial function 𝑓(𝑥) = 6 +
2𝑥2 − 5𝑥4.
2. Decide whether the function is a polynomial function. If so, write it is standard form and state its degree,
type, and leading coefficient. 𝑔(𝑥) = 𝜋𝑥4 + √6
Use direct substitution to evaluate the polynomial function for the given value of x. 3. 𝑓(𝑥) = 5𝑥3 − 2𝑥2 + 10𝑥 − 15; 𝑥 = −1 4. ℎ(𝑥) = 𝑥 +
1
2𝑥4 −
3
4𝑥3 + 10; 𝑥 = −4
Use synthetic substitution to evaluate the polynomial function for the given value of x. 5. 𝑔(𝑥) = 𝑥3 + 8𝑥2 − 7𝑥 + 35; 𝑥 = −6 6. ℎ(𝑥) = −7𝑥3 + 11𝑥2 + 4𝑥; 𝑥 = 3
Describe the degree and leading coefficient of the polynomial function whose graph is shown.
𝑥
𝑥
2
Algebra 2 5 Polynomials and Polynomial Functions Practice Problems
Page 2 of 12
7. Describe the end behavior of the graph of the polynomial function by completing these statements: 𝒇(𝒙) →
17. (5.5) Divide using polynomial long division (3𝑥2 − 11𝑥 − 26) ÷ (𝑥 − 5)
18. (5.4) Find the real-number solutions of the equation by factoring 𝑔3 + 3𝑔2 − 𝑔 − 3
19. (5.3) Simplify (𝑥 + 5)(𝑥 − 5)
20. (3.2) Solve the system of equations {3𝑥 + 𝑦 = 162𝑥 − 3𝑦 = −4
5.7 Apply the Fundamental Theorem of Algebra 1. Copy and complete: For the equation (𝑥 − 1)2(𝑥 + 2) = 0, a(n) _______ solution is 1 because the factor 𝑥 − 1
appears twice.
Identify the number of solutions or zeros. 2. 9𝑡6 − 14𝑡3 + 4𝑡 − 1 = 0 3. 𝑓(𝑥) = 16𝑥 − 22𝑥3 + 6𝑥6 + 19𝑥5 − 3
Find all the zeros of the polynomial function. 4. ℎ(𝑥) = 𝑥3 + 5𝑥2 − 4𝑥 − 20
5. 𝑓(𝑥) = 𝑥4 + 𝑥3 + 2𝑥2 + 4𝑥 − 8
6. 𝑔(𝑥) = 𝑥4 − 2𝑥3 − 3𝑥2 + 2𝑥 + 2
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.
7. −2, 1, 3
8. 3i, 2 – i
9. −4, 1, 2 − √6
Algebra 2 5 Polynomials and Polynomial Functions Practice Problems
Page 6 of 12
Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function.
Determine the numbers of positive real zeros, negative real zeros, and imaginary zeros for the function with the given degree and graph. Explain your reasoning.
14. Degree: 3
Word problem
15. From 1990 to 2003, the number N of inland lakes in Michigan infested with zebra mussels can be modeled
by the function
𝑁 = −0.028𝑡4 + 0.59𝑡3 − 2.5𝑡2 + 8.3𝑡 − 2.5
where t is the number of years since 1990. In which year did the number of infested inland lakes first reach
120?
Mixed Review 16. (5.6) List the possible rational zeros of the function. 𝑔(𝑥) = 𝑥3 − 4𝑥2 + 𝑥 − 10
17. (5.6) Find all the real zeros of 𝑔(𝑥) = 2𝑥3 − 7𝑥2 + 9
18. (5.5) Divide using polynomial long division (8𝑥2 + 34𝑥 − 1) ÷ (4𝑥 − 1)
19. (5.4) Factor 𝑥3 + 𝑥2 + 𝑥 + 1
20. (5.3) Simplify (3𝑥2 − 5) + (7𝑥2 − 3)
5.8 Analyze Graphs of Polynomial Functions Graph the function.
1. 𝑓(𝑥) = (𝑥 − 2)2(𝑥 + 1)
2. 𝑔(𝑥) =1
3(𝑥 − 5)(𝑥 + 2)(𝑥 − 3)
3. ℎ(𝑥) = 4(𝑥 + 1)(𝑥 + 2)(𝑥 − 1)
4. 𝑓(𝑥) = 2(𝑥 + 2)2(𝑥 + 4)2
5. 𝑔(𝑥) = (𝑥 − 3)(𝑥2 + 𝑥 + 1)
Describe and correct the error in graphing f. 6. 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 1)2
Estimate the coordinates of each turning point an dstate whether each corresponds to a local maximum or a local minimum. Then estimate all real zeros and determind the least degree the function can have.
Algebra 2 5 Polynomials and Polynomial Functions Practice Problems
Page 7 of 12
7.
8.
9.
10.
11. Which point is a local maximum of the function 𝑓(𝑥) = 0.25(𝑥 + 2)(𝑥 − 1)2?
A. (-2, 0) B. (-1, 1) C. (1, 0) D. (2, 1)
12. Why is the adjective local, used to describe the maximums and minimums of cubic functions, not required
for quadratic functions?
Graph the function. Then identify its domain and range. 13. 𝑓(𝑥) = 𝑥2(𝑥 − 2)(𝑥 − 4)(𝑥 − 5) 14. 𝑓(𝑥) = (𝑥 + 2)(𝑥 + 1)(𝑥 − 1)2(𝑥 − 2)2
Word problem 15. For a swimmer doing the breaststroke, the function
7. Max: (-0.3, 0.3), Min: (0.9, -1.3), Zeros: -0.75, 0, 1.4, Degree: 3 8. local maximum: (20.5, 22.4), local minimum: (1.5, 25.7); zero: 2.7, least degree: 3
9. local maximums: (1, 0), (3, 0), local minimum: (2, 22); zeros: 1, 3, least degree: 4 10. local maximums: (21.1, 0.8), (1.9, 8), local minimums: (22.2, 238), (0.3, 241), (2.8, 213); zeros: 22.6, 21.2, 1.5, 2.2, 3, least degree: 6 11. B 12. Cubic has no absolute max or min
13. , Domain: All real numbers, Range: All real numbers
14. ;
domain: all real numbers, range: y ≥
−21.3
Algebra 2 5 Polynomials and Polynomial Functions Practice Problems
10. 𝑓(𝑥) = −4𝑥2 + 15𝑥 11. 𝑓(𝑥) = −0.5𝑥3 + 5𝑥2 − 2.5𝑥 + 3 12. 5; 6; there must be one more data point than the degree of the equation. 13. 𝑑 = 0.5𝑛2 − 1.5𝑛