Download Algebra II Chapter 5 Test Review

1
Algebra II – Chapter 5 Test Review
Standards/Goals:
 F.IF.7.c: I can identify the degree of a polynomial function.
 F.1.a./A.APR.1.: I can evaluate and simplify polynomial expressions and equations.
 F.1.b./ A.SSE.2.: I can factor polynomials using a variety of methods (factor theorem, synthetic
division, long division, sums and differences of cubes, grouping)
 F.2.a./A.APR.2.: I can determine the number and type of rational zeros for a polynomial function.
 F.2.b./A.APR.3.: I can find all rational zeros of a polynomial function.
 F.2.c.: I can recognize the connection among zeros of a polynomial function, x-intercepts, factors of
polynomials, and solutions of polynomial equations.
 F.2.d./F.IF.7./F.IF.4.: I can use technology to graph a polynomial function and approximate the
zeros, minimum, and maximum; determine domain and range of the polynomial function.
 A.SSE.1a.: I can interpret expressions that represent a quantity in terms of its context (interpreting
parts of an expression, such as terms, factors, and coefficients.)
 A.SSE.1b.: I can interpret expressions that represent a quantity in terms of its context.
 A.APR.4.: I can prove polynomial identities and use them to describe numerical relationships.
 A.REI.11.: I can recognize and use function notation to represent polynomial equations.
 A.SSE.2 – Use structure of an expression to identify ways to rewrite.
o I can write a quadratic expression as the product of two binomials.
 A.APR.3 – Identify zeros of polynomials when suitable factorizations are available.
o I can solve a quadratic equation and identify the zeros using graphing and factoring
method.
 N.CN.7.: I can solve quadratic equations with real coefficients that have complex solutions.
 N.CN.8.: I can solve polynomial equation using the Fundamental Theorem of Algebra.
#1. Find all solutions of the equation:
Put answers in the box provided below and show ALL work in the space provided.
ANSWERS:
How many solutions should there be? ______
How many solutions are real? _______
How many solutions are imaginary? _______
What are the solutions? (List below):
) (
)
#2. Consider the factored polynomial function: f(x) (
Identify the zeros of this function and state the multiplicity of each.
a. _____________ is a zero of multiplicity _____________.
b. _____________ is a zero of multiplicity _____________.
c. State whether the graph crosses the x-axis or touches and turns around at each zero.
2
#3. Determine the degree, describe the behavior of the graph of the polynomial function as
(
) (
)(
)
and identify the x and y intercepts: f(x)
ANSWERS:
SHOW ALL WORK HERE:
Degree:
Behavior:
As x
f(x)
As x
f(x)
X-Intercepts:
________
________
________
Y-Intercepts:
#4. Use all available methods (in particular, the Conjugate Roots Theorem, if applicable) to
factor the following polynomial equation completely, making use of the given zero.
f(x) =
;
is a zero
Show work here:
ANSWER: ________________________________________________
3
#5. What are the real roots of:
?
#6. What is the factored form of:
?
#7. What are the zeros of: y = (x + 9)(4x + 2)(x – 8)?
#8. What is the multiplicity of f(x) = (
#9. For this polynomial:
intercept?
(
) ?
) , what type of behavior will the graph exhibit at the x-
#10. A quintic polynomial has rational coefficients. If √ and 10 – i are roots of P(x) = 0, what
are the additional roots?
#11. Factor the following and determine the zeros and the multiplicities.
( )
#12. A polynomial with real coefficients has roots of: 7, -8, 10i, and √ . What must be some
other ROOTS of this polynomial?
4
#13. The following is the graph of a cubic function. How many ‘distinct’ zeros does it have?
How many zeros are ‘real’ and how many are complex?
#14. Evaluate this function for x = -2: f(x) =
.
#15. A fourth degree polynomial, P(x) with real coefficients has 4 distinct zeros, 2 of them are
10 and 6 – i. What must be true about the other two?
#16. A fifth degree polynomial has been factored to be:
( ) (
)(
) (
) . How many x-intercepts does it have?
5
#17. What are the zeros of:
( )
(
)(
)(
)?
#18. Factor:
#19. The following polynomials have a degree of 5. How many ‘distinct’ real zeros are there for
each graph? How many are real and how many are complex/imaginary?
#1.
#2.
#3.
#20. What numbers are zeros and what are their multiplicities for:
6
#21. F.2.c.: Consider the graph of the function: f(x) =
, which has one xintercept at (-4,0). Find all the other zeros of the function algebraically. Show your work, and
explain how you found all your answer. Additionally, name the intervals where the function
both INCREASES and DECREASES.
7
FLASHBACK:
Solve, graph and write each in interval form:
#1. | |
#3.
| |
#2. | |
#4.
| |
#5. Consider the following quadratic:
Determine the number and type of roots for this equation using the discriminant.
8
#6. Consider the following function:
a. What is the vertex of the function?
b. What is the axis of symmetry?
c. Does the parabola have a minimum or a maximum and where at?
d. Where is the y-intercept?
#7. Let m = 5 – 7i and h = 2 + 9i
a. Find m – 6h
b. Find m · h
c. What is ?
d. What is m + h?
#8. Use the following functions to answer the questions:
f(x) = 3x – 2
g(x) =
h(x) = 4x + 8
a. Find ( )( ).
b. Find (g – k)(5)
k(x) =