Question

Which polynomial is factored completely?

Answer 1

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

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

Answer 2

Answer:

The polynomial which is factored completely is  $2g^2+5g+4$ as these are not further factored.        

Step-by-step explanation:

To find : Which polynomial is factored completely?

Solution :

We factor the given polynomial one by one,

1) $g^5-g$

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

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

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

2) $4g^3 + 18g^2+20g$

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

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

$4g^3 + 18g^2+20g=2g[g(2g+5)+2(2g+5)]$

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

3) $24g^2 - 6g^4$

$24g^2 - 6g^4=6g^{2}(4-g^{2})$

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

4) $2g^2+5g+4$

This expression is can not be factored with rational number as

$D=b^2-4ac\\D=5^2-4(2)(4)\\D=25-32\\D=-7$

Discriminant D<0 so there is no rational roots.

So, The polynomial which is factored completely is  $2g^2+5g+4$ as these are not further factored.