Polynomial Test Review
Name:___________________________________#
I can factor the difference of squares. I can solve quadratic equations by factoring.
1. a. Factor: ℎ(𝑥) = 16𝑥 2 − 9
b. Solve: 𝑧 2 − 19 = 81.
I can write a unique function that matches a parabola.
2. Write the equation for this graph.
I can identify key information easily obtainable from different forms of quadratic equations. I can summarize how to find the other information.
For each of the following quadratics, identify all of the key features.
3. First underline the information you can get from the function without using your calculator. Then identify the following for each equation.
a) 𝑓(𝑥) = (𝑥 + 3)2 − 25
b) 𝑓(𝑥) = 2𝑥 2 + 7𝑥 − 4
c) 𝑓(𝑥) = −2 (𝑥 + 1)(𝑥 − 3)
Parabola Opens:
Parabola Opens:
Parabola Opens:
Vertex:
Vertex:
Vertex:
Zeros:
Zeros:
Zeros:
Y-intercept:
Y-intercept:
Y-intercept:
Axis of Symmetry:
Axis of Symmetry:
Axis of Symmetry:
I can use key features to find possible equations for an unlabeled graph.
4. Determine which of the following equations might represent the below graph. Explain your choices.
a)
b)
c)
d)
𝑓(𝑥) = 2(𝑥 + 4)(𝑥 − 1.5)
𝑔(𝑥) = 𝑥 2 − 4𝑥 + 3
ℎ(𝑥) = 0.5(𝑥 − 2)2 − 3
𝑗(𝑥) = −(𝑥 − 3)(𝑥 + 1)
I can use the process called “completing the square” to rewrite a
quadratic trinomial in vertex form or as a perfect square trinomial.
I can solve a quadratic equation by completing the square.
5. Complete the square to find Vertex form:
a) 𝑓(𝑥) = 𝑥 2 − 10𝑥 + 3
b) Complete the square to solve:
0 = 𝑥 2 + 16𝑥 + 60
I can define the imaginary number i. I can calculate powers of i. I can add, subtract and multiply complex numbers. I understand that the
product of complex conjugates is a real number.
6. Simplify each of the following:
a) √−72
b) 𝑖17
c) 3𝑖(𝑖 − 4)
d) (5 − 2𝑖)(5 + 2𝑖)
I can determine the number and type of zeros of a quadratic function. I can solve quadratic equations with complex solutions.
7. Use the discriminant to determine the number and type of zeros for each of the following.
a) 𝑔(𝑥) = 2𝑥 2 + 4𝑥 − 3
b) 𝑣(𝑥) = 6𝑥 2 − 2𝑥 + 1
c) 𝑘(𝑥) = 4𝑥 2 + 4𝑥 + 1
8. Solve. Leave your answers in simplified exact form. 𝑣(𝑥) = 6𝑥 2 − 2𝑥 + 1
I can analyze a cubic function to determine domain and range in context.
9. An amusement park keeps track of the number of people in the park at any time. A regression equation describes the number of people in the
park, p(x). The variable x represents the time, in hours, after the park opens. 𝑝(𝑥) = 29.89𝑥 3 − 1017.89𝑥 2 + 8458.68𝑥 − 5167.8
a) Using the regression equation, how many people will be in the park after 6 hours?
b) State the domain and range of this function in the context of the problem.
I can determine if a graph or an equation represents an even or odd function.
10. Which of the following are odd? Explain your choices.
a)
b) 𝑓(𝑥) = 𝑥 5 − 7
c) 𝑔(𝑥) = 2𝑥 3 + 𝑥
I can find intervals of increase or decrease, end behaviors and possible number of zeros based on the degree of the polynomial.
1
11.Sketch the graph of 𝑓(𝑥) = (𝑥 − 1)(𝑥 − 3)(𝑥 + 2)2 without the use of a calculator. Explain how you graphed the function. Describe the
2
interval(s) of increase using the proper notation.
Polynomial Test Review
Name:___________________________________#
I can draw a graph of a polynomial function given the characteristics.
12. Emily is looking at a graph that has a local maximum at (-1,4) and local minimums at (-5,-6) and(3,-6).
a) Sketch the graph that shows the shape of a polynomial function with these characteristics.
b) What is the lowest possible degree of this polynomial? Explain .
I can write polynomials in standard and factored form. I can find the zeros of polynomials.
13. Find the zeros of the polynomial function then write it in standard form.
College prep: 𝑐(𝑥) = (𝑥 + 3)(𝑥 − 3)(2𝑥 + 1)
Honors: ℎ(𝑥) = (𝑥 − 2)2 (3𝑥 + 1)(3𝑥 − 1)
I can find key features of a polynomial function and sketch a graph by knowing the standard or factored form.
14. a) Without your calculator, write a polynomial equation in standard form with the following characteristics:
College prep: Zeros at x=5 and x=-4 and x = 0 going through the point (1,40).
Honors: Zeros at x=5 multiplicity 3 and x = 0 going through (-1, -108).
15. I can apply the fundamental theorem of algebra to polynomials with complex roots.
𝑓(𝑥) has two imaginary solutions and one real solution.
Compare 𝑔(𝑥) and 𝑓(𝑥). Which statement describes both polynomials? Both
functions have even degrees, both functions have odd degrees, or one function
has an even degree and the other has an odd degree. Explain your choice.
I can determine the number of relative and absolute extrema relative to the degree of a polynomial.
16. List the number of possible extrema for each polynomial.
a) 4th degree polynomial
b) 7th degree polynomial
I can divide a polynomial by a binomial using long division to determine if it is a factor..
17. Divide 18𝑤 4 + 7𝑤 2 + 4 𝑏𝑦 (3𝑤 − 1) to determine if it is a factor.
I can divide a polynomial by a binomial using synthetic division.
18. Divide 5𝑚5 + 9𝑚2 + 14 by (𝑚 + 1)using synthetic division. Write the dividend as the product of the divisor and the quotient plus the
remainder.
I can use the remainder theorem to evaluate a polynomial for a given value. I can use the remainder theorem to find a missing value.
𝑝(𝑥)
19. Given
= 𝑥 2 + 6𝑥 + 11 𝑅 − 4, find 𝑝(2).
(𝑥−2)
I can factor a sum of cubes and a difference of cubes.
20. a) Factor the sum of cubes. College prep: 𝑤 3 + 64
Honors: 8𝑥 3 + 27
I can find the solution to a polynomial inequality graphically and algebraically.
21. Solve by factoring and sketching. 𝑥 3 + 4𝑥 2 + 𝑥 − 6 > 0
I can find the solution to a polynomial inequality graphically. I can represent a solution to a polynomial inequality using inequalities, interval
notation, number lines, and verbal descriptions in context.
22. Emilio has been trying to regulate the pH level in his tropical fish aquarium for 5 hours. In order to make the water safe for his fish, Emilio
must keep the pH level less than or equal to 9. The function 𝑝(𝑥) = −0.34𝑥 3 + 2.652𝑥 2 − 5.4638𝑥 + 11.1114 represents the pH level in the tank x
hours since Emilio began to regulate it.
Write and solve an inequality to determine the time intervals during which the pH level was safe for Emilio’s fish. State your solution in interval
notation and explain in words what your solution means. Sketch the graph you used to solve.
Polynomial Test Review
Name:___________________________________#