Answer: 691
Explanation:
There are 3 different ways to find the remainder. I am not sure which method you are supposed to use, so I will solve using all 3 methods.
Long Division:
10x³ + 24x² + 77x + 230
x - 3 ) 10x⁴ - 6x³ + 5x² - x + 1
- (10x⁴ - 30x³) ↓ ↓ ↓
24x³ + 5x² ↓ ↓
- (24x³ - 72x²) ↓ ↓
77x² - x ↓
- (77x² - 231x) ↓
230x + 1
- (230x - 690)
691 ← remainder
Synthetic Division:
x - 3 = 0 ⇒ x = 3
3 | 10 -6 5 -1 1
| ↓ 30 72 231 690
10 24 77 230 691 ← remainder
Remainder Theorem:
f(x) = 10x⁴ - 6x³ + 5x² - x + 1
f(3) = 10(3)⁴ - 6(3)³ + 5(3)² - (3) + 1
= 810 - 162 + 45 - 3 + 1
= 691