Question

which polynomial is factored completely?
$g^5 - g$
$4g^3 + 18g^2 + 20g$
$24g^2 - 6g^4$
$2g^2 + 5g + 4$

Explanation:

Step1: Factor first polynomial

$g^5 - g = g(g^4 - 1) = g(g^2-1)(g^2+1) = g(g-1)(g+1)(g^2+1)$

Step2: Factor second polynomial

$4g^3 + 18g^2 + 20g = 2g(2g^2 + 9g + 10) = 2g(2g+5)(g+2)$

Step3: Factor third polynomial

$24g^2 - 6g^4 = 6g^2(4 - g^2) = 6g^2(2-g)(2+g)$

Step4: Check fourth polynomial

For $2g^2+5g+4$, discriminant $\Delta = 5^2 - 4(2)(4) = 25 - 32 = -7 < 0$. No real factors, so it is fully factored over real numbers.

Answer:

$2g^2 + 5g + 4$