Answer: 3
<u>Step-by-step explanation:</u>
(x + 4i) is a root which means its conjugate (x - 4i) is also a root
(x + 4i)(x - 4i) = x² + 16
Use Long Division for 2x⁵ - 13x⁴ + 22x³ - 187x² - 160x + 336 ÷ x² + 16
<u> 2x³ - 13x² - 10x + 21 </u>
x² + 16 ) 2x⁵ - 13x⁴ + 22x³ - 187x² - 160x + 336
- <u>(2x⁵ + 0x⁴ + 32x³) </u> ↓ ↓ ↓
-13x⁴ - 10x³ - 187x² ↓ ↓
- <u>(-13x⁴ +0x³ - 208x²) </u> ↓ ↓
-10x³ + 21x² - 160x ↓
- <u>(-10x³ + 0x² - 160x)</u> ↓
21x² + 0x + 336
- <u>(21x² + 0x + 336)</u>
0
Use Synthetic Division to divide (x - 7) by the reduced polynomial (2x³ - 13x² - 10x + 21)
7 | 2 -13 -10 21
<u>| ↓ 14 7 -21 </u>
2 1 -3 0 ← remainder is 0
Use grouping to factor the reduced polynomial (2x² + x - 3)
2x² - 2x + 3x - 3
= 2x(x - 1) +1(x - 1)
= (2x + 1)(x - 1)
The factors are: (x - 7) (2x + 1) (x - 1) (x + 4i) (x - 4i)
The first 3 are real roots and the last 2 are complex roots
