Download CA ADV Algebra Standard 06

CA ADV Algebra Standard 06
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
____
____
____
1. Find the complex conjugate of
.
a.
b.
c.
d.
2. Add. Write the result in the form
.
(6 – 8 ) + (–7 – 3 )
a. 13 – 5
b. –1 – 11
c. –2 – 10
d. 3 – 15
3. Subtract. Write the result in the form
.
(–3 – 9 ) – (2 – 5 )
a. –1 – 14
b. –5 – 4
c. 2 – 11
d. 6 + 7
4. Find
by graphing on the complex plane.
imag
a.
10
8
6
4
2
–10 –8
–6
–4
–2
–2
–4
–6
–8
–10
Sum: –7 – 4i
2
4
6
8
real
imag
b.
10
8
6
4
2
–10 –8
–6
–4
–2
–2
2
4
6
8
real
4
6
8
real
4
6
8
real
–4
–6
–8
–10
Sum: 1 + 8i
imag
c.
10
8
6
4
2
–10 –8
–6
–4
–2
–2
2
–4
–6
–8
–10
Sum: –7 + 4i
imag
d.
10
8
6
4
2
–10 –8
–6
–4
–2
–2
2
–4
–6
–8
–10
____
Sum: 4 – 7i
5. Multiply
a.
b.
c.
. Write the result in the form
.
d.
____
6. Simplify
.
a.  2 – 8
17
17
2
2
b.  –
5
c. 
2
17
d. 
2
5
3
8
17
+
+
i
2
3
____
7. What expression is equivalent to
a.
b.
c.
d.
?
____
8. Write the simplest polynomial function with the zeros
a.
b.
c.
d.
,
, and
Numeric Response
9. What value of d makes the equation
true?
.
CA ADV Algebra Standard 06
Answer Section
MULTIPLE CHOICE
1. ANS: D
Rewrite as
Find
.
Simplify.
=
=
=
.
Feedback
A
B
C
D
This is the number in a + bi form. Now find the complex conjugate.
You changed the sign of both the real and imaginary parts. Only change the sign of the
real part.
You changed the sign of the real part. Only change the sign of the imaginary part.
Correct!
PTS: 1
DIF: Basic
REF: Page 352
OBJ: 5-5.5 Finding Complex Conjugates
STA: 2A6.0
TOP: 5-5 Complex Numbers and Roots
2. ANS: B
To add complex numbers, add the real parts and the imaginary parts. To subtract complex numbers, subtract
the real parts and the imaginary parts.
(6 – 8 ) + (–7 – 3 ) = –1 – 11
Feedback
A
B
C
D
Check whether you should add or subtract the two complex numbers.
Correct!
Add real parts and imaginary parts.
Add real parts and imaginary parts.
PTS: 1
DIF: Average
REF: Page 383
OBJ: 5-9.3 Adding and Subtracting Complex Numbers
STA: 2A6.0
TOP: 5-9 Operations with Complex Numbers
3. ANS: B
To add complex numbers, add the real parts and the imaginary parts. To subtract complex numbers, subtract
the real parts and the imaginary parts.
(–3 – 9 ) – (2 – 5 ) = –5 – 4
Feedback
A
B
C
D
Check whether you should add or subtract the two complex numbers.
Correct!
Subtract real parts and imaginary parts.
Subtract real parts and imaginary parts.
PTS: 1
DIF: Average
REF: Page 383
OBJ: 5-9.3 Adding and Subtracting Complex Numbers
TOP: 5-9 Operations with Complex Numbers
STA: 2A6.0
4. ANS: C
Graph
and
on the complex plane. Connect each of these numbers to the origin with a line
segment. Draw a parallelogram that has these two line segments as sides. The vertex that is opposite the
origin represents the sum of the two complex numbers,
. Therefore,
=
.
Feedback
A
B
C
D
Check the sign of the imaginary parts of the numbers.
Plot each term, and then create a parallelogram using the two points and the origin. The
sum is the final point opposite the origin.
Correct!
The real axis is the x-axis, and the imaginary axis is the y-axis.
PTS:
OBJ:
TOP:
KEY:
5. ANS:
1
DIF: Average
REF: Page 384
5-9.4 Adding Complex Numbers on the Complex Plane
STA: 2A6.0
5-9 Operations with Complex Numbers
complex number operations | graph complex numbers | graphing complex numbers | complex plane
B
Distribute.
Use
Write in
.
form.
Feedback
A
B
C
D
Use the Distributive Property. Then simplify by using the fact that i squared is equal to
–1.
Correct!
Use the Distributive Property. Then simplify by using the fact that i squared is equal to
–1.
Use the Distributive Property. Then simplify by using the fact that i squared is equal to
–1.
PTS:
OBJ:
TOP:
6. ANS:
1
DIF: Basic
REF: Page 384
5-9.5 Multiplying Complex Numbers
5-9 Operations with Complex Numbers
C
=
Multiply by the conjugate.
=
Distribute.
=
Use
.
STA: 2A6.0
 17 +
2
8
17
i
Simplify.
Feedback
A
B
C
D
Remember that i^2 = –1
Remember that i^2 = –1
Correct!
Remember that i^2 = –1
PTS: 1
STA: 2A6.0
7. ANS: D
=
=
=
=
DIF: Average
REF: Page 385
OBJ: 5-9.7 Dividing Complex Numbers
TOP: 5-9 Operations with Complex Numbers
Multiply.
Combine like terms.
Simplify.
.
Feedback
A
B
C
D
First, expand the square and multiply. Then, combine like terms and simplify.
i squared is equal to –1.
First, expand the square and multiply. Then, combine like terms and simplify.
Correct!
PTS: 1
DIF: Advanced
STA: 2A6.0
TOP: 5-9 Operations with Complex Numbers
8. ANS: A
There are five roots:
,
,
,
, and . (By the Irrational Root Theorem and Complex
Conjugate Root Theorem, irrational and complex roots come in conjugate pairs.) Since it has 5 roots, the
polynomial must have degree 5.
Write the equation in factored form, and then multiply to get standard form.
Feedback
A
B
C
D
Correct!
i squared is equal to –1, so the opposite is equal to 1.
–4x(–5) = 20x
Only the irrational roots and the complex roots come in conjugate pairs. There are five
roots in total.
PTS: 1
DIF: Average
REF: Page 447
OBJ: 6-6.3 Writing a Polynomial Function with Complex Zeros
STA: 2A6.0
TOP: 6-6 Fundamental Theorem of Algebra
NUMERIC RESPONSE
9. ANS: 6
PTS: 1
DIF: Average
STA: 2A6.0
TOP: 5-5 Complex Numbers and Roots