Download Algebra II Benchmark 2 Review Systems of 3 x 3 equations 1

Algebra II
Benchmark 2 Review
Systems of 3 x 3 equations
1) Algebraically solve the following systems of equations for x, y, and z.
4 x  y  3z  11
2x  3y  2z  9
x  y  z  3
2) Algebraically solve the following systems of equations for x, y, and z.
x  6 y  2z  9
3 x  2 y  3 z  1
5x  5 y  2 z  7
Solve the following by Completing the Square
3) x 2  10 x  21  0
4) 3x 2  6 x  1  0
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Algebra II
Benchmark 2 Review
Composite Functions
5) Given: f ( x )  
1
( x  5)  3 find f 1 ( x )
4
x 1
1
and g ( x )  x  4
2
3
find f (g (6))
7) Given f ( x ) 
6) Given f ( x )  3( 2 x  5) find f
1
( x)
8) Given f ( x )  2 x  3 and g ( x )  x 2  1
find g ( f ( 2))
Radical equations: Solve for all possible roots of x:
9)
11)
x 2 x
10)
x 4 8  7
9  2 x  8  1
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Algebra II
Benchmark 2 Review
Factor completely or Find the product:
12) 6a 2  9ab  3b  2a
13) 3x 4  6 x 3  27 x 2  54 x
14) x ( x  5)( x  3)
15) ( x  2)( x  2)( x  3)
16) ( x  2)( x  2) 2 ( x  3)
Rational Equations: For #17 – 19, find the solution set
4x
12
2
x3
17) x  3
y
8
1
 
18) y  1 y y  1
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Algebra II
3x
12
 2
2
19) x  1 x  1
Benchmark 2 Review
2x
3

20) x  4 x  1
Quadratic - Linear equations: Algebraically determine the values for x and y that
satisfy questions #21 – 23 below.
21)
23)
y  x2  8
y  5  2x
22)
y   x 2  6x  3
y  x 1
y  2x2  4x  5
3x  y  1
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Algebra II
Benchmark 2 Review
long vs. Synthetic Division
Note: when using synthetic division and the divisor has a coefficient of the x term, divide your quotient by
that number (not the remainder).
24) Determine the quotient and the remainder when you divide (2 x 4  9 x 3  21x 2  26 x  12) by ( 2 x  3) .
25) Determine the quotient and the remainder of
26) Simplify:
2x3  2x  9
2x  4
15 x 3  x 2  12 x  12
3x  2
Imaginary Numbers / Roots:
For #27 - #32 be sure your answer is in simplest a  bi form.
27) Find the product of (9  i )( 5  3i )
28) Find the product of (3  2i ) and (7  6i )
29) Find the area of the rectangle shown:
4i
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Algebra II
30) Solve x 2  4 x  22
Benchmark 2 Review
31) Solve for all values of x:  2 x  4  ( x  3) 2
32) Solve for x and express in simplest a  bi form by completing the square: 3x 2  6 x  4  0 .
Higher Order Equations
33) Use any method of your choice to prove that x  5 is a factor of x 4  3x 3  7 x 2  11x  20 .
Explain your method of proof.
34) Without using a calculator, show that x+3 is a factor of the function f ( x )  x 3  2 x 2  2 x  3
Explain your answer.
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Algebra II
Benchmark 2 Review
35) For question #35, state the roots, state the factors & write an equation for the following graphs
(a)
(b)
Roots _________________
Factors _________________
Equation: ________________
(c)
Roots _________________
Roots _________________
Factors _________________
Factors _________________
Equation: ________________
Equation: ________________
(d)
(e)
Roots _________________
Roots _________________
Factors _________________
Factors _________________
Equation: ________________
Equation: ________________
(f) Looking at graphs 35d and 35e, what is the same about the graphs?
What is the difference between about the graphs? How do you make this happen?
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Algebra II
Benchmark 2 Review
Calculator Work: For questions #36 - #38, write down all steps used on the calculator.
36) To the nearest tenth, determine the value of x that satisfies 2 x  2 x  11
37) If f ( x )  4 x 2  4 and h ( x ) 
1
x  3 , find to the nearest tenth, the value that satisfies f ( x )  h ( x ) .
2
38) If p( x )  x 3  5 and q( x )  2 x  1 , find to the nearest tenth, the value that satisfies p( x )  q( x ) .
Trig functions: For questions #39 - #41 simplify each trigonometric function to a single term.
39) sin 2  csc  sec 
41)
cos  csc 
[1]
cot 
Unit Circle
43) If the terminal side of angle  , in standard position,
passes through the point (-5, -12) what is the
numerical value of each of the following:
40) csc   cos  cot 
42) 1  tan 2 
44) If the terminal side of angle  , in standard position,
passes through the point (8, -6) what is the numerical
value of each of the following:
sin 
sin 
tan 
tan 
cos 
cos 
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