When factored completely, [tex]m^5 + m^3 -6m[/tex] is equivalent to [tex]m(m-\sqrt2)(m+\sqrt2)(m^2+3)[/tex]
To factorize the polynomial
[tex]m^5 + m^3 -6m[/tex]
notice that the GCF of each term is m. So we can factorize m out of each term as follows
[tex]m^5 + m^3 -6m=m(m^4 + m^2 -6)[/tex]
the fourth-degree polynomial in the brackets an be treated as a quadratic polynomial
[tex](m^2)^2+(m^2)-6[/tex]
and factorized as such. The quadratic has the constant term [tex]-6[/tex]. We have to look for factors of [tex]-6[/tex] that add up to [tex]+1[/tex](the co-efficient of the [tex]m^2[/tex] term).
The factors that satisfy the condition are [tex]-2[/tex] and [tex]+3[/tex]. So we can factorize the fourth-degree polynomial as
[tex](m^2)^2+(m^2)-6=(m^2-2)(m^2+3)[/tex]
We have factorized the polynomial so far as
[tex]m^5 + m^3 -6m=m(m^2-2)(m^2+3)[/tex]
But the factor [tex](m^2-2)[/tex] is a difference of two squares and can be factorized as
[tex]m^2-2=(m-\sqrt2)(m+\sqrt2)[/tex]
Our final factored polynomial is
[tex]m(m-\sqrt2)(m+\sqrt2)(m^2+3)[/tex]
Learn more about polynomial factorization here https://brainly.com/question/16078564
