Factor the polynomial completely, or state that the polynomial is prime: 5x⁴ − 80.

Step-by-step solution

1. 1

   Take out the numerical GCF. Both terms share a factor of $5$: $5x^4 - 80 = 5(x^4 - 16)$. Doing this first exposes the structure hidden by the coefficients.

2. 2

   Spot the difference of squares. $x^4 = (x^2)^2$ and $16 = 4^2$, so $x^4 - 16 = (x^2 - 4)(x^2 + 4)$.

3. 3

   Apply the pattern a second time. The first bracket is itself a difference of squares: $x^2 - 4 = (x - 2)(x + 2)$.

4. 4

   Decide whether x² + 4 can be split. A sum of squares has no real roots, so over the real numbers $x^2 + 4$ is irreducible and the factoring stops here.

5. 5

   State the complete factorisation. Combining everything gives $5(x - 2)(x + 2)(x^2 + 4)$.

6. 6

   Verify. $(x-2)(x+2)(x^2+4) = (x^2-4)(x^2+4) = x^4 - 16$, and $5(x^4 - 16) = 5x^4 - 80$.