1) f(x) = 5x⁴ - 36x² - 81
- Complex zeros: 4
- Real zeros: 2 (possible), Imaginary zeros: 2 (possible)
- Positive zeros: 1 (possible)
- Negative zeros: 1 (possible)
- Possible rational zeros: ±1, ±3, ±9, ±27, ±81, ±1/5, ±3/5, ±9/5, ±27/5, ±81/5
2) f(x) = 15x⁵ + 3x⁴ + 140x³ + 28x² + 45x + 9
- Complex zeros: 5
- Real zeros: 1 or 3 or 5 (possible), Imaginary zeros: 0 or 2 or 4 (possible)
- Positive zeros: 0 (possible)
- Negative zeros: 1 or 3 or 5 (possible)
- Possible rational zeros: ±1, ±3, ±9, ±1/3, ±1/5, ±3/5, ±9/5, ±1/15, ±3/15, ±9/15
3) f(x) = 5x³ - x² - 5x + 1
- Complex zeros: 3
- Real zeros: 1 or 3 (possible), Imaginary zeros: 0 or 2 (possible)
- Positive zeros: 1 or 3 (possible)
- Negative zeros: 0 or 2 (possible)
- Possible rational zeros: ±1, ±1/5
4) f(x) = 3x³ + 11x² + 5x - 3
- Complex zeros: 3
- Real zeros: 1 or 3 (possible), Imaginary zeros: 0 or 2 (possible)
- Positive zeros: 1 (possible)
- Negative zeros: 0 or 2 (possible)
- Possible rational zeros: ±1, ±3, ±1/3
5) f(x) = 10x⁵ - 15x⁴ + 12x³ - 18x² + 2x - 3
- Complex zeros: 5
- Real zeros: 1 or 3 or 5 (possible), Imaginary zeros: 0 or 2 or 4 (possible)
- Positive zeros: 1 or 3 or 5 (possible)
- Negative zeros: 0 (possible)
- Possible rational zeros: ±1, ±3, ±1/2, ±3/2, ±1/5, ±3/5, ±1/10, ±3/10
6) f(x) = 5x⁵ - 25x⁴ + 46x³ - 230x² + 9x - 45
- Complex zeros: 5
- Real zeros: 1 or 3 or 5 (possible), Imaginary zeros: 0 or 2 or 4 (possible)
- Positive zeros: 1 or 3 or 5 (possible)
- Negative zeros: 0 (possible)
- Possible rational zeros: ±1, ±3, ±5, ±9, ±15, ±45, ±1/5, ±3/5, ±9/5, ±15/5, ±45/5
7) f(x) = 2x⁴ - 11x² + 9
- Possible rational zeros: ±1, ±3, ±9, ±1/2, ±3/2, ±9/2
- Interval for real zeros: [-3, 3]
- Factored form: (2x² - 9)(x² - 1) = (√2x - 3)(√2x + 3)(x - 1)(x + 1)
8) f(x) = 27x³ + 1
- Possible rational zeros: ±1, ±1/3, ±1/9, ±1/27
- Interval for real zeros: [-1, 0]
- Factored form: (3x + 1)(9x² - 3x + 1)
Parent Tip: Review the logic above to help your child master the concept of fundamental theorem of algebra worksheet.
