Precalculus Complex Numbers and Complex Zeros Test Solutions and Scoring (PDF)

Precalculus math test solutions: Complex Numbers and Complex Zeros – Maths PDF

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This precalculus complex numbers test answers 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.

Worked solutions and scoring notes for the 9-question complex numbers test. Use the scoring line under each solution to award partial credit.

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 test answers: worked solutions

Question 1 – Powers and radicals [2 pts]

\(43=4\cdot10+3\), so \(i^{43}=-i\). \(\sqrt{-48}=i\sqrt{16\cdot3}=4i\sqrt{3}\).

\(-i\) and \(4i\sqrt{3}\)

Scoring: 1 point for each.

Question 2 – Multiply [2 pts]

\(3+12i-2i-8i^{2}=3+10i+8\).

\(11+10i\)

Scoring: 1 point for the expansion, 1 point for replacing i squared.

Question 3 – Divide [2 pts]

Multiply by \(\frac{2+3i}{2+3i}\): numerator \(10+15i+2i+3i^{2}=7+17i\); denominator \(4+9=13\).

\(\frac{7}{13}+\frac{17}{13}i\)

Scoring: 1 point for the conjugate step, 1 point for the result.

Question 4 – Modulus and conjugate [2 pts]

\(|z|=\sqrt{25+144}=13\). \(z\overline{z}=25+144=169\).

\(13\) and \(169\)

Scoring: 1 point for each.

Question 5 – Complex plane [3 pts]

The sum is the fourth corner of the parallelogram.

(a) \(z_{1}=2+3i\), \(z_{2}=-1+i\). (b) \(1+4i\). (c) \(\sqrt{4+9}=\sqrt{13}\).

\(2+3i,\ -1+i\); \(1+4i\); \(\sqrt{13}\)

Scoring: 1 point for each part.

Question 6 – Quadratic with complex roots [2 pts]

Lowest point (2, 16).

\(x=\frac{4\pm\sqrt{16-80}}{2}=\frac{4\pm 8i}{2}\). The graph confirms there are no real zeros.

\(x=2\pm 4i\)

Scoring: 1 point for the discriminant, 1 point for the roots.

Question 7 – Classify the roots [2 pts]

\(36-40=-4< 0\), two non-real roots. \(x=\frac{-6\pm 2i}{2}\).

\(-4\); \(x=-3\pm i\)

Scoring: 1 point for the discriminant and classification, 1 point for the roots.

Question 8 – Conjugate pairs and degree [3 pts]

(a) \(3+i\). (b) \((x-3+i)(x-3-i)=(x-3)^{2}+1=x^{2}-6x+10\). (c) \((x-1)(x-2)=x^{2}-3x+2\); times \(x^{2}-6x+10\) gives \(x^{4}-9x^{3}+30x^{2}-42x+20\).

\(3+i\); \(x^{2}-6x+10\); \(x^{4}-9x^{3}+30x^{2}-42x+20\)

Scoring: 1 point for each part.

Question 9 – Verify a root [2 pts]

\((2-i)^{2}=4-4i+i^{2}=3-4i\). Then \(3-4i-4(2-i)+5=3-4i-8+4i+5=0\).

The value is 0

Scoring: 1 point for squaring correctly, 1 point for showing the sum is 0.

Total score

Total

20 points

Key facts behind this precalculus complex numbers test answers

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 test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All Precalculus chapters: Precalculus math.

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