Key takeaway: This algebra 2 real complex numbers test has nine questions that mix classification, plotting, the imaginary unit, addition, multiplication, powers of \(i\), arrows in the plane and a solution check. Write each result as \(a+bi\) and show your steps. No calculator is needed, so keep exact forms.
This test has 9 questions worth 20 points in all. Allow 45 minutes. Show all your work: partial credit is given for correct steps. Write complex numbers in the form a + bi and simplify every power of i. A calculator is not needed.
- Grade: Algebra 2
- Chapter: Real and Complex Numbers
- Common Core: N-RN.A.1, N-RN.A.2, N-CN.A.1, N-CN.A.2
- Time: 45 minutes
- Total: 20 points (Question 1: 2 pts; Question 2: 2 pts; Question 3: 2 pts; Question 4: 2 pts; Question 5: 3 pts; Question 6: 2 pts; Question 7: 3 pts; Question 8: 2 pts; Question 9: 2 pts)
Algebra 2 Real Complex Numbers Test: questions 1–5
1. Classify marked numbers (2 points)
Two numbers are marked on the number line.
- Is each number rational or irrational? (1 point)
- Name the smallest of these sets that contains each number: natural, integer, rational, irrational. (1 point)
2. Points in the complex plane (2 points)
Write the complex number for each point shown.
- Point M (1 point)
- Point N (1 point)
3. Imaginary unit (2 points)
- Simplify \(\sqrt{-72}\). (1 point)
- Solve \(3x^{2}+48=0\). (1 point)
4. Add and subtract (2 points)
Write each result in the form \(a+bi\).
- \((5-3i)+(-8+7i)\) (1 point)
- \((6+2i)-(9-5i)\) (1 point)
5. Multiply (3 points)
Write each product in the form \(a+bi\).
- \((4-i)(2+3i)\) (1 point)
- \((3+5i)^{2}\) (1 point)
- \((7+2i)(7-2i)\) (1 point)
Algebra 2 Real Complex Numbers Test: questions 6–9
6. Powers of i (2 points)
Simplify.
- \(i^{38}\) (1 point)
- \(i^{83}\) (1 point)
7. Arrows in the plane (3 points)
The arrows show two complex numbers \(z_1\) and \(z_2\).
- Write \(z_1\) and \(z_2\) in the form \(a+bi\). (1 point)
- Find \(z_1+z_2\). (1 point)
- Find \(z_1-z_2\). (1 point)
8. Check a solution (2 points)
Show that \(2+i\) is a solution of \(x^{2}-4x+5=0\).
9. Find the error (2 points)
Liam writes \((2+3i)(2-3i)=4-9i^{2}=4-9=-5\). Find his mistake and give the correct product.
What do the opening questions ask?
The first decides whether two marked numbers are rational or irrational. The second reads two points off the complex plane, so state the real part first. Then simplify \(\sqrt{-72}\) and solve \(3x^{2}+48=0\). Remember to pull the perfect square out of the radical.
How should you handle the operations?
For sums and differences, handle the two kinds of part one at a time. For products, expand with the distributive property and then replace \(i^{2}\). The conjugate pair \((7+2i)(7-2i)\) has a fast answer, because the middle terms cancel. Powers such as \(i^{38}\) and \(i^{83}\) reduce by dividing the exponent by 4 and using the remainder.
What do the last three questions test?
The arrows question asks you to write two numbers from a picture and then add and subtract them. Next, check that \(2+i\) solves \(x^{2}-4x+5=0\) by substitution. Finally, find Liam’s mistake in \((2+3i)(2-3i)\), where a sign slips when \(i^{2}\) is replaced.
What should you do before you submit?
Reread every answer and make sure no \(i^{2}\) remains. Then confirm that each result has the form \(a+bi\) with both parts clearly written. Once you grade the paper with the key, list the type of slip that cost points.
See every chapter on the Algebra 2 math page.
