Question

When factored completely, m^5 + m^3 -6m is equivalent to what

Answer 1

When factored completely, $m^5 + m^3 -6m$ is equivalent to $m(m-\sqrt2)(m+\sqrt2)(m^2+3)$

To factorize the polynomial

$m^5 + m^3 -6m$

notice that the GCF of each term is m. So we can factorize m out of each term as follows

$m^5 + m^3 -6m=m(m^4 + m^2 -6)$

the fourth-degree polynomial in the brackets an be treated as a quadratic polynomial

$(m^2)^2+(m^2)-6$

and factorized as such. The quadratic has the constant term $-6$. We have to look for factors of $-6$ that add up to  $+1$(the co-efficient of the $m^2$ term).

The factors that satisfy the condition are $-2$ and $+3$. So we can factorize the fourth-degree polynomial as

$(m^2)^2+(m^2)-6=(m^2-2)(m^2+3)$

We have factorized the polynomial so far as

$m^5 + m^3 -6m=m(m^2-2)(m^2+3)$

But the factor $(m^2-2)$ is a difference of two squares and can be factorized as

$m^2-2=(m-\sqrt2)(m+\sqrt2)$

Our final factored polynomial is

$m(m-\sqrt2)(m+\sqrt2)(m^2+3)$

Learn more about polynomial factorization here https://brainly.com/question/16078564