# Which polynomial is prime?

x^{3} + 3x^{2} - 2x - 6, x^{3} - 2x^{2} + 3x - 6, 4x^{4} + 4x^{3} - 2x - 2, 2x^{4} + x^{3} - x + 2

**Solution:**

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

Solving each polynomial one by one.

**a) x ^{3} + 3x^{2} - 2x - 6,**

x²(x + 3) -2(x + 3)

(x² - 2)(x + 3)

(x² - 2)(x + 3) are the factors ,

**therefore x ^{3} + 3x^{2} - 2x - 6 is not a prime polynomial.**

**b) x ^{3} - 2x^{2} + 3x - 6**

x²(x - 2) + 3(x - 2)

(x² + 3)(x - 2)

(x² + 3)(x - 2) are the factors,

**therefore x ^{3} - 2x^{2} + 3x - 6 in not a prime polynomial.**

c) 4x^{4} + 4x^{3} - 2x - 2

4x³ (x + 1) - 2(x + 1)

(4x³ - 2)(x + 1)

(4x³ - 2)(x + 1) are the factors,

**therefore 4x ^{4} + 4x^{3} - 2x - 2 is not a prime polynomial.**

d) 2x^{4} + x^{3} - 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 2x^{4} + x³ - x + 2 is the only prime polynomial from the given polynomials.

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