Complex Factoring Sphere Volume Worksheet Answer Key – Precalculus

Problem 1 Powers of i

Powers of i repeat every 4. 27 ÷ 4 leaves remainder 3, so i²⁷ = i³ = −i. 46 ÷ 4 leaves remainder 2, so i⁴⁶ = i² = −1.

Answer: i²⁷ = −i and i⁴⁶ = −1

Problem 2 Expanding a Product

(x + 5i)(x − 5i) = x² − 25i² = x² + 25.

Answer: x² + 25

Problem 3 A Sum of Squares

49 = 7², so x² + 49 = x² + 7² = (x + 7i)(x − 7i).

Answer: (x + 7i)(x − 7i)

Problem 4 A Non-Square Constant

√18 = 3√2, so x² + 18 = (x + 3√2i)(x − 3√2i).

Answer: (x + 3√2i)(x − 3√2i)

Problem 5 Solving a Sum of Squares

x² = −36, so x = ±√(−36) = ±6i. These are the two points on the imaginary axis.

Answer: x = 6i or x = −6i

Problem 6 Product of Conjugates

(3 + 2i)(3 − 2i) = 9 − 4i² = 9 + 4 = 13, which equals 3² + 2² = 13.

Answer: 13

Problem 7 A Leading Coefficient

4x² + 25 = (2x)² + 5² = (2x + 5i)(2x − 5i).

Check: (2x)² − (5i)² = 4x² + 25.

Answer: (2x + 5i)(2x − 5i)

Problem 8 Two Variables

9x² + 16y² = (3x)² + (4y)² = (3x + 4yi)(3x − 4yi).

Answer: (3x + 4yi)(3x − 4yi)

Problem 9 A Fourth-Degree Factorization

x⁴ − 81 = (x² − 9)(x² + 9) = (x − 3)(x + 3)(x + 3i)(x − 3i).

The zeros are 3, −3, 3i and −3i.

Answer: (x − 3)(x + 3)(x + 3i)(x − 3i)

Problem 10 Factor Out First

Divide by 2: x² + 49 = 0, so x = ±7i. The factored form is 2(x + 7i)(x − 7i).

Answer: x = ±7i; 2(x + 7i)(x − 7i)

Problem 11 A Quadratic with Complex Zeros

The discriminant is 100 − 136 = −36, so √(−36) = 6i. Then x = −10 ± 6i2 = −5 ± 3i.

The factored form is (x + 5 − 3i)(x + 5 + 3i).

Check: (x + 5)² − (3i)² = x² + 10x + 25 + 9.

Answer: x = −5 ± 3i

Problem 12 Completing the Square

x² − 4x + 4 = −29 + 4, so (x − 2)² = −25. Then x − 2 = ±5i and x = 2 ± 5i.

Answer: x = 2 + 5i and x = 2 − 5i

Problem 13 The General Identity

(a + bi)(a − bi) = a² − abi + abi − b²i² = a² − b²(−1) = a² + b².

With a = 7 and b = 4: (7 + 4i)(7 − 4i) = 49 + 16 = 65.

Answer: 65

Problem 14 Volume of a Sphere

V = 43πr³ = 43π(216) = 288π cubic inches.

288π ≈ 904.8.

Answer: 288π ≈ 904.8 cubic inches

Problem 15 Cavalieri with a Hemisphere

(a) Hemisphere slice: radius √(25 − 9) = 4, area 16π. Cylinder-minus-cone slice: ring with outer radius 5 and inner radius 3, area 25π − 9π = 16π. The areas match.

(b) The cylinder has volume π(25)(5) = 125π. The cone has volume 13π(25)(5) = 125π3. The hemisphere is 125π − 125π3 = 250π3, and the sphere is 500π3 ≈ 523.6.

Answer: both slices have area 16π; hemisphere 250π3, sphere 500π3 ≈ 523.6

Problem 16 Spherical Water Tank

(a) The radius is 6 ft. V = 43π(6)³ = 288π ≈ 904.8 cubic feet.

(b) 904.78 × 7.48 ≈ 6,768 gallons.

Answer: about 904.8 cubic feet, or about 6,768 gallons

Problem 17 Cone with a Dome

Cone: 13π(3²)(8) = 24π. Hemisphere: 23π(3³) = 18π.

Total = 24π + 18π = 42π ≈ 131.9 cubic centimeters.

Answer: 42π ≈ 131.9 cm³

Problem 18 A Quartic Equation

Treat it as a quadratic in x²: x⁴ + 5x² − 36 = (x² + 9)(x² − 4), since 9 · (−4) = −36 and 9 + (−4) = 5.

Then x² = −9 gives x = ±3i, and x² = 4 gives x = ±2.

Answer: x = 2, −2, 3i, −3i

Problem 19 Factoring a Cubic

Group: x²(x − 4) + 9(x − 4) = (x − 4)(x² + 9).

Then x² + 9 = (x + 3i)(x − 3i), so the factorization is (x − 4)(x + 3i)(x − 3i).

Answer: zeros 4, 3i and −3i

Problem 20 Building a Polynomial

Complex zeros of a real polynomial come in conjugate pairs, so 3 − i is also a zero.

(x − 3 − i)(x − 3 + i) = (x − 3)² − i² = x² − 6x + 10.

Multiply by (x − 2): x³ − 6x² + 10x − 2x² + 12x − 20 = x³ − 8x² + 22x − 20.

Check at x = 2: 8 − 32 + 44 − 20 = 0.

Answer: x³ − 8x² + 22x − 20