Complex Factorization and Volume of a Sphere: Precalculus Math Test Answer Key – PDF Download

Precalculus Complex Identities Test Answer Key: questions 1–5

Question 1: Powers of i (2 points)

1. \(38=36+2\), so \(i^{38}=i^{2}=-1\).
2. \(55=52+3\), so \(i^{55}=i^{3}=-i\).

Scoring: 1 point per part.

Question 2: Factor a sum of squares (2 points)

\(25x^{2}+4=(5x)^{2}+2^{2}=(5x+2i)(5x-2i)\). Check: \((5x+2i)(5x-2i)=25x^{2}-4i^{2}=25x^{2}+4\).

Scoring: 1 point for the factorization; 1 point for the expansion with \(i^{2}=-1\).

Question 3: Solve over the complex numbers (2 points)

1. \(x^{2}=-64\), so \(x=\pm 8i\).
2. \(x^{2}=-18\), so \(x=\pm i\sqrt{18}=\pm 3i\sqrt{2}\).

Scoring: 1 point per part (accept \(\pm i\sqrt{18}\) for part b).

Question 4: Fourth-degree factorization (3 points)

\(x^{4}-256=(x^{2}-16)(x^{2}+16)\). Then \(x^{2}-16=(x-4)(x+4)\) and \(x^{2}+16=(x+4i)(x-4i)\). \((x-4)(x+4)(x+4i)(x-4i)\)

Scoring: 1 point for the difference of squares; 1 point for the real factors; 1 point for the complex factors.

Question 5: Quadratic with no real zeros (3 points)

1. The vertex is \((3, 16)\), and the parabola opens upward, so it never reaches the x-axis. There is no real solution of \(x^{2}-6x+25=0\).
2. \(x^{2}-6x+25=(x^{2}-6x+9)+16=(x-3)^{2}+4^{2}=(x-3+4i)(x-3-4i)\). The zeros are \(3\pm 4i\).

Scoring: part (a) 1 point; part (b) 1 point for completing the square, 1 point for the factors.

Precalculus Complex Identities Test Answer Key: questions 6–8

Question 6: Matching cross-sections (3 points)

1. The radius is \(\sqrt{25-9}=4\), so the area is \(16\pi\) square ft.
2. The ring has area \(25\pi-9\pi=16\pi\) square ft.
3. The two solids have equal volumes.

Scoring: 1 point per part.

Question 7: Volume from a cylinder and a cone (2 points)

1. \(\pi(36)(6)-\frac{1}{3}\pi(36)(6)=216\pi-72\pi=144\pi\) cubic in.
2. The hemisphere has the same volume, \(144\pi\), so the sphere has \(2(144\pi)=288\pi\) cubic in.

Scoring: 1 point per part.

Question 8: Spherical tank (3 points)

1. The radius is \(10\). \(V=\frac{4}{3}\pi(1000)=\frac{4000\pi}{3}\) cubic ft.
2. \(\frac{4000\pi}{3}\approx 4188.8\) cubic ft.
3. \(4188.79 \times 7.48\approx 31332\) gallons.

Scoring: 1 point per part (accept 31,331 or 31,332).

Scoring summary

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Precalculus Complex Factorization and Volume of a Sphere test answer key

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