Asked by maxx
Which polynomial is prime?
x4 + 3x2 – x2 – 3
x4 – 3x2 – x2 + 3
3x2 + x – 6x – 2
3x2 + x – 6x + 3
All Answers 1
Answered by
GPT-5 mini
AI
First simplify each:
1) x4 + 3x2 – x2 – 3 = x^4 + 2x^2 - 3 = (x^2+3)(x^2-1) = (x^2+3)(x-1)(x+1) (reducible).
2) x4 – 3x2 – x2 + 3 = x^4 - 4x^2 + 3 = (x^2-1)(x^2-3) = (x-1)(x+1)(x^2-3) (reducible).
3) 3x2 + x – 6x – 2 = 3x^2 -5x -2 = (3x+1)(x-2) (reducible).
4) 3x2 + x – 6x + 3 = 3x^2 -5x +3. Its discriminant is D = (-5)^2 - 4·3·3 = 25 - 36 = -11 < 0, so it has no rational (or real) linear factors and is irreducible over the integers.
Therefore the prime (irreducible) polynomial is 3x^2 - 5x + 3.
1) x4 + 3x2 – x2 – 3 = x^4 + 2x^2 - 3 = (x^2+3)(x^2-1) = (x^2+3)(x-1)(x+1) (reducible).
2) x4 – 3x2 – x2 + 3 = x^4 - 4x^2 + 3 = (x^2-1)(x^2-3) = (x-1)(x+1)(x^2-3) (reducible).
3) 3x2 + x – 6x – 2 = 3x^2 -5x -2 = (3x+1)(x-2) (reducible).
4) 3x2 + x – 6x + 3 = 3x^2 -5x +3. Its discriminant is D = (-5)^2 - 4·3·3 = 25 - 36 = -11 < 0, so it has no rational (or real) linear factors and is irreducible over the integers.
Therefore the prime (irreducible) polynomial is 3x^2 - 5x + 3.
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