What must be subtracted from 2x⁴ – 5x³ + 5x + 40 so that the result is divisible by 2x – 1?
Answers
Given : 2x⁴ – 5x³ + 5x + 40
To find : What must be subtracted so that the result is divisible by 2x – 1
Solution:
f(x) = 2x⁴ – 5x³ + 5x + 40
Remainder Theorem
the result is divisible by 2x – 1 if f(1/2) = 0
f(1/2) = 2 (1/2)⁴ - 5(1/2)³ + 5(1/2) + 40
=> f(1/2) = 1/8 - 5/8 + 5/2 + 40
=> f(1/2) = 42
Hence 42 has to be subtracted from 2x⁴ – 5x³ + 5x + 40 so that the result is divisible by 2x – 1
Another method :
Division method
x³ - 2x² - x + 2
2x - 1 _| 2x⁴ – 5x³ + 5x + 40 |_
2x⁴ – x³
__________
– 4x³ + 5x + 40
– 4x³ + 2x²
___________
- 2x² + 5x + 40
- 2x² + x
_____________
4x + 40
4x - 2
________
42
Remainder = 42
Hence 42 must be subtracted from 2x⁴ – 5x³ + 5x + 40 so that the result is divisible by 2x – 1
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