Answer:
The polynomial which is factored completely is [tex]2g^2+5g+4[/tex] 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) [tex]g^5-g[/tex]
[tex]g^5-g=g(g^4-1)[/tex]
[tex]g^5-g=g(g^2-1)(g^2+1)[/tex]
[tex]g^5-g=g(g-1)(g+1)(g^2+1)[/tex]
2) [tex]4g^3 + 18g^2+20g[/tex]
[tex]4g^3 + 18g^2+20g=2g(2g^{2}+9g+10)[/tex]
[tex]4g^3 + 18g^2+20g=2g(2g^{2}+5g+4g+10)[/tex]
[tex]4g^3 + 18g^2+20g=2g[g(2g+5)+2(2g+5)][/tex]
[tex]4g^3 + 18g^2+20g=2g(2g+5)(g+2)[/tex]
3) [tex]24g^2 - 6g^4[/tex]
[tex]24g^2 - 6g^4=6g^{2}(4-g^{2})[/tex]
[tex]24g^2 - 6g^4=6g^{2}(2-g)(2+g)[/tex]
4) [tex]2g^2+5g+4[/tex]
This expression is can not be factored with rational number as
[tex]D=b^2-4ac\\D=5^2-4(2)(4)\\D=25-32\\D=-7[/tex]
Discriminant D<0 so there is no rational roots.
So, The polynomial which is factored completely is [tex]2g^2+5g+4[/tex] as these are not further factored.
