← Back to the Complex Factoring Sphere Volume test
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: N-CN.C.8, G-GMD.A.2 · Level: Grade 12 Precalculus
Question 1 Powers of i (2 points)
38 ÷ 4 leaves remainder 2, so i³⁸ = i² = −1. 55 ÷ 4 leaves remainder 3, so i⁵⁵ = i³ = −i.
Answer: −1 and −i
Scoring: 1 point for each power.
Question 2 Factoring Sums of Squares (2 points)
(a) x² + 8² = (x + 8i)(x − 8i).
(b) (5x)² + 3² = (5x + 3i)(5x − 3i).
Answer: (a) (x + 8i)(x − 8i); (b) (5x + 3i)(5x − 3i)
Scoring: 1 point for each part.
Question 3 A Radical Solution (2 points)
x² = −75, so x = ±√75 i = ±5√3 i.
Answer: x = ±5√3 i
Scoring: 1 point for the setup x² = −75, 1 point for the simplified solutions.
Question 4 A Fourth-Degree Polynomial (3 points)
(a) x⁴ − 625 = (x² − 25)(x² + 25) = (x − 5)(x + 5)(x² + 25).
(b) (x − 5)(x + 5)(x + 5i)(x − 5i).
(c) The points are 5, −5, 5i and −5i.
Answer: (x − 5)(x + 5)(x + 5i)(x − 5i)
Scoring: 1 point for each part.
Question 5 Complex Zeros of a Quadratic (3 points)
(a) 64 − 100 = −36.
(b) x = 8 ± √(−36)2 = 8 ± 6i2 = 4 ± 3i.
(c) (x − 4 − 3i)(x − 4 + 3i).
Answer: x = 4 ± 3i
Scoring: 1 point for each part.
Question 6 Cavalieri’s Principle (3 points)
(a) The slice is a disk of radius √(36 − 16) = √20, with area 20π.
(b) The slice is a ring with outer radius 6 and inner radius 4, with area 36π − 16π = 20π.
(c) The slices match at every height and the solids have equal heights, so the two solids have equal volumes.
Answer: 20π and 20π; the volumes are equal
Scoring: 1 point for each part.
Question 7 Volume of a Sphere (3 points)
The radius is 9 cm. V = 43π(9)³ = 43π(729) = 972π.
972π ≈ 3,053.6 cubic centimeters.
Answer: 972π ≈ 3,053.6 cm³
Scoring: 1 point for the radius, 1 point for the formula with values, 1 point for the result.
Question 8 Building a Cubic (2 points)
The conjugate 2 − 3i is also a zero. (x − 2 − 3i)(x − 2 + 3i) = (x − 2)² + 9 = x² − 4x + 13.
Multiply by (x − 1): x³ − 4x² + 13x − x² + 4x − 13 = x³ − 5x² + 17x − 13.
Answer: x³ − 5x² + 17x − 13
Scoring: 1 point for the conjugate pair factor, 1 point for the product.
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