Algebra 2 – Unit 5 Test Review Worksheet
10 pt HW
Name_________________________ Block _____Assigned _____ Due ______
Simplify the expression (5.3) Make sure you use the ____________ property.
This is the _________ or _____________ of two polynomials.
(6w3 + 2w2  3w  l) + (5w3 + 9w  8)
(4m4  m2 + 5m)  (2m3 + m2  2m + 6)
This is the _________________ of two polynomials.
(2x3 + x)(x4 + 3x3  2x2 +1)
(2p + l)(6p2  p + 8)
Factor theorem: (5.5 p. 365)
Make sure you know factors, roots, and zeros
A polynomial f(x) has a factor of x-k if and only if f (k) = 0
Thus: A polynomial with a factor of x+3 means f (-3 ) = 0, -3 is a root, -3 is a zero
Your turn:
Factor of x + 3 = 0, x = ____ f (_____) = 0, _____ is a root of f(x)
meaning y = 0 and the point is on the x axis, “zero”
factor of (x – 4) = 0 , x = ____ f(_____) = 0, ___ is a root of f(x), ___ is a zero
factor of (x + 5) = 0 , x = ____ f(_____) = 0, ___ is a root of f(x), ___ is a zero
(5.7)Write a polynomial function f of least degree that has rational coefficients, a
leading coefficient of 1, and the given zeros. Write the following as ________, and
________
1, 2, 6
1, 2, 5
5, 0, 2i, 2i
1, ,
13
44
(5.7)Use the calculator to find the roots of the function. 𝑥 3 − 𝑥 2 − 8𝑥 + 5
First, graph the equation in the _____ screen. 2nd ______, _______, left, right, guess.
There must be a sign change with the _____ in order to calculate the zer0. Round to the
hundredths
(5.7) Determine the possible number of possible zeros for a function: p. 379
Determine the ______ of the function. Write the function from highest degree _____ to _____.
−6𝑥 4 − 8𝑥 + 7𝑥 2 + 𝑥 5 − 3𝑥 3 + 1
Is this in order of the highest power left to right? ______ if not, rewrite it.
Number of zeros? ________
(5.3, 5.4) Factor
Go to page 354 Special Factoring patterns. Make sure you know!
Sum of Two Cubes
Difference of Two Cubes
_____________________________
______________________________
x3  512
7h3 +448
Factor by grouping.
z3  2z2  16z  32
12x3  6x2 + 2x  1
Difference of squares
z3  z2 + 5z  5
𝑥 4 − 169
Match the polynomial with its factorization.
______ 3x2 + 11x + 6
A. 2x3(x + 2)(x  2)(x2 + 3)
______ x3  4x2 + 4x – 16
B. 2x(x + 4)(x  4)
______ 125x3 – 216
C. (3x + 2)(x + 3)
______2x7  2x5  24x3
D. (x2 + 4)(x  4)
______2x5 + 4𝑥 4  4x3  8x2
E. 2x2(x2  2)(x + 2)
______2x3  32x
F. (5x  6)(25x2 + 30x + 36)
Show work here:
Factor and solve:
2𝑥 4 − 13𝑥 2 − 45=0
𝑚3 + 6𝑚2 − 4𝑚 − 24 = 0
(5.5)Divide using polynomial long division.
(5𝑥 4 − 2𝑥 3 − 7𝑥 2 − 39) ÷ (𝑥 2 + 2𝑥 − 4)
(3𝑥 3 + 11𝑥 2 + 4𝑥 + 1) ÷ (𝑥 2 + 𝑥)
Divide using synthetic division.
X+5 is a factor so the zero is ____
(𝑥 4 + 4𝑥 3 + 16𝑥 − 35) ÷ (𝑥 + 5)
_____
(5.5)
Zeros are -2, -2/3,+4
Given the polynomial function and a zero, find the other zeros.. use the example above
𝑓(𝑥) = 𝑥 3 − 2𝑥 2 − 21𝑥 − 18 𝑧𝑒𝑟𝑜 𝑖𝑠 − 3 𝑠𝑜 𝑓𝑎𝑐𝑡𝑜𝑟 𝑖𝑠 (
(x
Use synthetic division:
) is a factor so the zero is - 3
Given the polynomial
-3
𝑓(𝑥) = 𝑥 3 − 2𝑥 2 − 40𝑥 − 64;
If x – 8 is a factor, x = ______
Keep factoring until done
What are the solutions?
𝑥 − 8 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟
)
p
q

factor of Constant term
0
factor of leading coefficient
(5.6)Find all real zeros of f( X ) = X 3  7 X 2 + 14 X 8.
STEP 1 List the possible rational zeros. The leading coefficient is 1 and the constant term is 8. The
possible rational zeros are: x = ±, 1, ±2, ±4, ±8
STEP 2 Test these zeros using synthetic division.
Test x = 4: Factor would be ( x- 4)
4
1
1
7
14
8
4
12
8
3
2
0
4 is a zero
Because 4 is a zero of f, write f(x) = (x  4)(x2 3x + 2).
STEP 3 Factor the trinomial and use the factor theorem.
f(x) = (x  4)(x 2  3x + 2) = (x  4)(x  2)(x  1)
The zeros are 1, 2, and 4.
(5.6) List all possible zeros and find all zeros of the function applying the rational zero theorem:
2𝑥 4 + 6𝑥 3 − 7𝑥 + 9
p = ______ q = ________
𝑝
𝑞
From your 5.2 notes: Max number of turns:
Max Number of turns – Take the highest power and subtract 1 from it this is the max number of possible
turns. Determine what the end behaviors are and the max number of turns.
y   x 3  x 2  3x  3
h(x) = 2x9  8x7 + 7x5
𝑦 = −3𝑥 6 + 2𝑥 5 − 4𝑥
𝑦 = 2𝑥 4 + 3𝑥 7 − 3
As x increases ______
As x increases ______
As x decreases ______
As x decreases ______
f  x   ____ as x  
f  x   ____ as x  
f  x   ____ as x  
f  x   ____ as x  
Describe the end behavior of each graph, odd or even, number of real zeros
(5.7) Find all the zeroes of
𝑥 4 − 6𝑥 3 + 7𝑥 2 + 6𝑥 − 8
( Find your p and q. Try 1 or -1 first)
Use synthetic division and keep going until you can factor.