Complex Numbers Assessment
Algebra II / Precalculus
Complex Numbers & Operations
Total Score ______ / 30 pts
Name:
Date:
Period:
Instructions: Simplify each expression into standard form \(a + bi\) (where \(a, b \in \mathbb{R}\)). Recall \(i = \sqrt{-1}\) and \(i^2 = -1\). Show all algebraic steps clearly in the workspace provided.
Part 1: Addition & Subtraction
11 Points
1
Find the sum and express in standard form: \( (-5 + 8i) + (12 - 14i) \)
[3 pts]
Answer:
2
Find the difference and simplify completely: \( (9 - 4i) - (15 - 11i) \)
[4 pts]
Answer:
3
Perform the indicated operations with multiple terms: \( (6 + 3i) + (2 - 7i) - (-4 + 5i) \)
[4 pts]
Answer:
Complex Numbers Mastery Assessment Page 1 of 2
Assessment • Part 2
Multiplication & Multi-Step Operations
Name: 19 Points Total
4
Distribute and simplify completely: \( 4i(7 - 3i) \)
[4 pts]
Answer:
5
Multiply the two binomials and express in standard form: \( (3 + 5i)(2 - 4i) \)
[5 pts]
Answer:
6
Multiply the complex conjugate pair: \( (5 - 6i)(5 + 6i) \)
[5 pts]
Classification (Real or Non-real):
Product:
7
Multi-Step: Expand and simplify completely in standard form: \( 2(4 - 3i) - (1 + 2i)^2 \)
[5 pts]
Final Standard Form:
Complex Numbers Mastery Assessment Page 2 of 2
Complex Numbers Answer Key
Teacher Guide & Scoring Rubric
Complex Numbers Assessment — Answer Key
Total Points 30 Points
Scoring Standards: Award partial credit for correct algebraic setups. Full credit requires standard form \(a + bi\) with completely simplified real and imaginary components.
Scale: A (27-30) • B (24-26) • C (21-23)
Part 1: Addition & Subtraction Solutions
Questions 1–3 • 11 Points
1
\( (-5 + 8i) + (12 - 14i) \)
3 Points
Step-by-Step Solution:
\( = (-5 + 12) + (8i - 14i) \)
\( = 7 + (-6i) \)
\( = 7 - 6i \)
Watch For: Students adding real to imaginary (e.g., \(-5 + 8i = 3i\)).
Point Breakdown:
- Group terms: +1 pt
- Real part (7): +1 pt
- Imag part (\(-6i\)): +1 pt
Ans: \( 7 - 6i \)
2
\( (9 - 4i) - (15 - 11i) \)
4 Points
Step-by-Step Solution:
Distribute negative: \( = 9 - 4i - 15 + 11i \)
Group like terms: \( = (9 - 15) + (-4i + 11i) \)
\( = -6 + 7i \)
Watch For: Failing to distribute negative to \(-11i\), resulting in \(-15i\).
Point Breakdown:
- Distribute minus: +1 pt
- Real part (\(-6\)): +1 pt
- Imag part (\(+7i\)): +1 pt
- Standard form: +1 pt
Ans: \( -6 + 7i \)
3
\( (6 + 3i) + (2 - 7i) - (-4 + 5i) \)
4 Points
Step-by-Step Solution:
Remove parentheses & distribute: \( = 6 + 3i + 2 - 7i + 4 - 5i \)
Combine real terms: \( 6 + 2 + 4 = 12 \)
Combine imaginary terms: \( (3 - 7 - 5)i = -9i \)
\( = 12 - 9i \)
Point Breakdown:
- Distribute \(-(-4+5i)\): +1 pt
- Real sum (12): +1 pt
- Imag sum (\(-9i\)): +1 pt
- Standard form: +1 pt
Ans: \( 12 - 9i \)
Complex Numbers Mastery • Scoring Guide Page 1 of 2
Answer Key • Part 2
Multiplication & Multi-Step Operations Solutions
Questions 4–7 • 19 Points
4
\( 4i(7 - 3i) \)
4 Points
Distribute: \( = 28i - 12i^2 \)
Apply \(i^2 = -1\): \( = 28i - 12(-1) = 28i + 12 \)
Standard form \(a + bi\): \( = 12 + 28i \)
Note: Deduct 1 pt if written as \(28i + 12\) without standard \(a+bi\) order.
Rubric:
• Distribution: 1 pt
• \(i^2 = -1\) conversion: 1 pt
• Correct standard form: 2 pts
Ans: \( 12 + 28i \)
5
\( (3 + 5i)(2 - 4i) \)
5 Points
FOIL: \( = 3(2) + 3(-4i) + 5i(2) + 5i(-4i) \)
\( = 6 - 12i + 10i - 20i^2 \)
Replace \(i^2 = -1\): \( = 6 - 2i - 20(-1) = 6 - 2i + 20 \)
\( = 26 - 2i \)
Rubric:
• 4 FOIL terms: 2 pts
• \(i^2 = -1\) conversion: 1 pt
• Combine like terms: 1 pt
• Final standard form: 1 pt
Ans: \( 26 - 2i \)
6
\( (5 - 6i)(5 + 6i) \)
5 Points
Difference of squares: \( = 5^2 - (6i)^2 = 25 - 36i^2 \)
Substitute \(i^2 = -1\): \( = 25 - 36(-1) = 25 + 36 = 61 \)
Standard form: \( 61 \) (or \(61 + 0i\))
Classification: Real number (zero imaginary component)
Rubric:
• Expansion method: 2 pts
• \(i^2 = -1\) substitution: 1 pt
• Product = 61: 1 pt
• Real classification: 1 pt
Ans: \( 61 \), Real
7
\( 2(4 - 3i) - (1 + 2i)^2 \)
5 Points
Part A (Distribute): \( 2(4 - 3i) = 8 - 6i \)
Part B (Expand \((1+2i)^2\)): \( 1 + 4i + 4i^2 = 1 + 4i - 4 = -3 + 4i \)
Part C (Subtract): \( (8 - 6i) - (-3 + 4i) = 8 - 6i + 3 - 4i \)
Combine: \( = (8 + 3) + (-6i - 4i) = 11 - 10i \)
Rubric:
• \(8 - 6i\): 1 pt
• Square expansion: 2 pts
• Correct subtraction: 1 pt
• Final \(11 - 10i\): 1 pt
Ans: \( 11 - 10i \)
Complex Numbers Mastery • Scoring Guide Page 2 of 2
