By Contributor

Updated Aug 30, 2022

In algebra, a prime polynomial (also called an irreducible polynomial) cannot be factored further over the integers. Recognizing these polynomials is essential before declaring a problem unsolvable.

Step 1: Check for a Greatest Common Factor

Begin by factoring out any common monomial factor from every term. If none exists, move to the next step.

Step 2: Apply Special Factorization Formulas

Test the standard identities:

• Difference of squares: a² – b² = (a – b)(a + b)
• Perfect square trinomials: (a ± b)² = a² ± 2ab + b²

Step 3: Factor a Quadratic with Coefficient 1

For a monic quadratic x² + Bx + C, look for two integers whose product is C and sum is B. If no such pair exists, the polynomial is likely prime.

Step 4: Factor a General Quadratic

For Ax² + Bx + C, compute the discriminant D = B² – 4AC. If D is not a perfect square, the quadratic has no rational roots and is irreducible over the integers.

Step 5: Exhaust All Possibilities

Only after checking GCF, special formulas, and the discriminant should you conclude that the polynomial is prime.

Step 6: Example – x² + 2x + 8

Assume a factorization of the form (x + a)(x + b). Then ab = 8 and a + b = 2. The integer pairs for 8 are (1,8) and (2,4), but neither sums to 2. The discriminant is 4 – 32 = –28, not a perfect square, confirming irreducibility.

Step 7: Declare the Polynomial Prime

After verifying that no common factor exists and that all standard factorization methods fail, you can confidently state that the polynomial is prime.