Which polynomial is prime?
x3 + 3x2 - 2x - 6, x3 - 2x2 + 3x - 6, 4x4 + 4x3 - 2x - 2, 2x4 + x3 - x + 2

Solution:

A prime polynomial is a polynomial which cannot be further factorised.

Solving each polynomial one by one.

a) x3 + 3x2 - 2x - 6,

x²(x + 3) -2(x + 3)

(x² - 2)(x + 3)

(x² - 2)(x + 3) are the factors ,

therefore x3 + 3x2 - 2x - 6 is not a prime polynomial.

b) x3 - 2x2 + 3x - 6

x²(x - 2) + 3(x - 2)

(x² + 3)(x - 2)

(x² + 3)(x - 2) are the factors,

therefore x3 - 2x2 + 3x - 6 in not a prime polynomial.

c) 4x4 + 4x3 - 2x - 2

4x³ (x + 1) - 2(x + 1)

(4x³ - 2)(x + 1)

(4x³ - 2)(x + 1) are the factors,

therefore 4x4 + 4x3 - 2x - 2 is not a prime polynomial.

d) 2x4 + x3 - x + 2

This polynomial cannot be further factorised ,therefore it is a prime polynomial.

Which polynomial is prime? x³ + 3x² - 2x - 6, x³ - 2x² + 3x - 6, 4x⁴ + 4x³ - 2x - 2, 2x⁴ + x³ - x + 2

Summary:

A prime polynomial is a polynomial which cannot be further factorised. So by factorising each polynomial we got that 2x4 + x³ - x + 2 is the only prime polynomial from the given polynomials.

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