Question

Which polynomial is factored completely?

g^5 - g

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

24g^2 - 6g^4

2g2 + 5g + 4

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

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

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.