Complex Factoring Sphere Volume Test – Precalculus Math (Printable PDF)

Question 1 Powers of i (2 points)

Evaluate i³⁸ and i⁵⁵.

Question 2 Factoring Sums of Squares (2 points)

Factor over the complex numbers.

1. x² + 64 (1 point)
2. 25x² + 9 (1 point)

Question 3 A Radical Solution (2 points)

Solve x² + 75 = 0. Simplify the radical.

Question 4 A Fourth-Degree Polynomial (3 points)

The figure plots the four zeros of x⁴ − 625.

1. Factor x⁴ − 625 completely over the real numbers. (1 point)
2. Factor it completely over the complex numbers. (1 point)
3. Label the four plotted points with their values. (1 point)

Question 5 Complex Zeros of a Quadratic (3 points)

Consider x² − 8x + 25 = 0.

1. Find the discriminant. (1 point)
2. Solve the equation. (1 point)
3. Write the quadratic in factored form over the complex numbers. (1 point)

Question 6 Cavalieri’s Principle (3 points)

A hemisphere of radius 6 and a cylinder of radius 6 and height 6 with an upside-down cone removed are cut at height 4.

1. Find the area of the hemisphere’s slice. (1 point)
2. Find the area of the cylinder-minus-cone slice. (1 point)
3. State what Cavalieri’s principle says about the volumes. (1 point)

Question 7 Volume of a Sphere (3 points)

A spherical globe has a diameter of 18 centimeters. Find its volume exactly and to the nearest tenth.

Question 8 Building a Cubic (2 points)

Write a cubic polynomial with real coefficients that has zeros 1 and 2 + 3i. Give it in standard form.