Precalculus Complex Numbers and Complex Zeros Test with Point Values (PDF)
This precalculus complex numbers test for Precalculus covers complex numbers and complex zeros (Common Core N-CN.A.1, N-CN.A.2, N-CN.A.3, N-CN.C.7, N-CN.C.8, N-CN.C.9), with a free PDF to print.
Time and points
- Grade: Precalculus
- Chapter: Complex Numbers and Complex Zeros
- Standards: N-CN.A.1, N-CN.A.2, N-CN.A.3, N-CN.C.7, N-CN.C.8, N-CN.C.9
- Time: 45 minutes
- Total: 20 points
Precalculus complex numbers test: questions
Question 1 – Powers and radicals [2 pts]
Simplify \(i^{43}\) and \(\sqrt{-48}\).
Question 2 – Multiply [2 pts]
Multiply: \((3-2i)(1+4i)\).
Question 3 – Divide [2 pts]
Write \(\frac{5+i}{2-3i}\) in the form \(a+bi\).
Question 4 – Modulus and conjugate [2 pts]
For \(z=-5+12i\), find \(|z|\) and \(z\overline{z}\), where \(\overline{z}\) is the conjugate.
Question 5 – Complex plane [3 pts]
The graph shows \(z_{1}\) and \(z_{2}\). (a) Write each as \(a+bi\). (b) Find \(z_{1}+z_{2}\). (c) Find \(|z_{1}|\).
Question 6 – Quadratic with complex roots [2 pts]
Solve \(x^{2}-4x+20=0\).
Question 7 – Classify the roots [2 pts]
Find the discriminant of \(x^{2}+6x+10=0\), classify its roots, and solve.
Question 8 – Conjugate pairs and degree [3 pts]
A degree-4 polynomial \(f\) with real coefficients and leading coefficient 1 has zeros \(1\), \(2\) and \(3-i\). (a) Find the fourth zero. (b) Write the quadratic factor with real coefficients. (c) Write \(f(x)\) in standard form.
Question 9 – Verify a root [2 pts]
Show that \(2-i\) is a root of \(x^{2}-4x+5\).
Key facts behind this precalculus complex numbers test
- i^2 = -1, and powers of i repeat in a cycle of 4: i, -1, -i, 1.
- The conjugate of a + bi is a – bi, and (a + bi)(a – bi) = a^2 + b^2.
- Modulus: |a + bi| = sqrt(a^2 + b^2), the distance from the origin in the complex plane.
- A negative discriminant gives two complex roots that are conjugates.
- Fundamental Theorem of Algebra: a degree-n polynomial has n complex zeros counting multiplicity.
How to use this precalculus complex numbers test
Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.
All Precalculus chapters: Precalculus math.
