Precalculus Complex Numbers and Complex Zeros Worksheet (PDF)

Precalculus math worksheet: Complex Numbers and Complex Zeros – Maths PDF

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This precalculus complex numbers worksheet 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.

These 20 problems practice complex numbers and complex zeros (Common Core N-CN.A.1 to N-CN.A.3 and N-CN.C.7 to N-CN.C.9). You will compute with i, find moduli and conjugates, solve quadratics and find all zeros of polynomials.

Precalculus · Common Core State 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

Precalculus complex numbers worksheet: exercises 1 to 10

Exercise 1 – Square roots of negatives

Write \(\sqrt{-36}\) and \(\sqrt{-20}\) in terms of \(i\).

Exercise 2 – Powers of i

Cycle of powers of i.

Simplify \(i^{27}\) and \(i^{50}\).

Exercise 3 – Add complex numbers

Add: \((4+3i)+(2-5i)\).

Exercise 4 – Subtract complex numbers

Subtract: \((7-2i)-(3-6i)\).

Exercise 5 – Multiply

Multiply: \((2+3i)(4-i)\).

Exercise 6 – Square a complex number

Simplify \((3+4i)^{2}\).

Exercise 7 – Conjugate product

Find the conjugate of \(5-12i\) and the product of \(5-12i\) with its conjugate.

Exercise 8 – Divide

Write \(\frac{3+2i}{1-i}\) in the form \(a+bi\).

Exercise 9 – Divide a real number

Write \(\frac{10}{3-i}\) in the form \(a+bi\).

Exercise 10 – Modulus

Find \(|5-12i|\) and \(|-3+4i|\).

Exercises 11 and up

Exercise 11 – Read the complex plane

Four complex numbers.

Write the complex numbers \(A\), \(B\), \(C\) and \(D\) shown, and find the modulus of \(A\).

Exercise 12 – A pure imaginary solution

Solve \(x^{2}+4=0\).

Exercise 13 – Complex roots by the formula

Graph of y = x^2 – 6x + 13.

Solve \(x^{2}-6x+13=0\) and explain why the graph has no \(x\)-intercepts.

Exercise 14 – Complex roots with a leading coefficient

Solve \(2x^{2}+2x+5=0\).

Exercise 15 – Discriminant

A parabola with no x-intercepts.

Use the discriminant to tell how many real and non-real roots each has: (a) \(x^{2}-4x+4\); (b) \(x^{2}+x+3\); (c) \(x^{2}-5x+2\).

Exercise 16 – Conjugate pair

A polynomial with real coefficients has the zero \(2+3i\). Name another zero and write the quadratic factor with real coefficients.

Exercise 17 – All zeros of a cubic

Find all zeros of \(f(x)=x^{3}-3x^{2}+7x-5\), given that \(x=1\) is a zero.

Exercise 18 – Missing zeros

A polynomial of degree 5 with real coefficients has zeros \(2\), \(1+i\) and \(-3i\). Find the other two zeros.

Exercise 19 – Impedance

Impedances in the complex plane.

In an electric circuit, two parts have impedances \(Z_{1}=3+4i\) ohms and \(Z_{2}=2-i\) ohms in series. Find \(Z_{1}+Z_{2}\) and its magnitude to the nearest hundredth. If the current is \(I=2+i\) amps through \(Z_{1}\), find the voltage \(V=IZ_{1}\).

Exercise 20 – Verify a root

Show that \(1+2i\) is a root of \(x^{2}-2x+5\).

Exercise 21 – Power of a complex number

Show that \((1+i)^{2}=2i\), then simplify \((1+i)^{4}\).

Key facts behind this precalculus complex numbers worksheet

Key facts

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the precalculus complex numbers worksheet as a PDF with the button below and keep it with your notes.

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