what must be added to p(x)=6x4+8x3+18x2+20x+5 so that the resulting polynomial is exactly divisible by g(x)=3x2+2x+1
Answers
Given : p(x) = 6x⁴ + 8x³ + 18x² + 20x + 5 , g(x) = 3x² + 2x + 1
To find : what must be added to p(x) so that the resulting polynomial is exactly divisible by g(x)
Solution:
p(x) = 6x⁴ + 8x³ + 18x² + 20x + 5
g(x) = 3x² + 2x + 1
p(x) = g(x)q(x) + r(x)
2x² + 4x/3 + 40/9
3x² + 2x + 1 _| 6x⁴ + 8x³ + 18x² + 20x + 5 |_
6x⁴ + 4x³ + 2x²
______________
4x³ + 16x² + 20x + 5
4x³ + 8x²/3 + 4x/3
___________________
40x²/3 + 56x/3 + 5
40x²/3 + 80x/9 + 40/9
__________________
424x/9 + 5/9
Remainder = (424x + 5)/9
Remainder has to be sutraced hence - Remainder to be added
Hence - (424x + 5)/9 must be added to p(x) = 6x⁴ + 8x³ + 18x² + 20x + 5 so that the resulting polynomial is exactly divisible by g(x) = 3x² + 2x + 1
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