Chapter 4.4, Problem 8E

To determine

To factor: the polynomial completely.

Answer to Problem 8E

The factorized form is $2m^6-24m^5+64m^4=2m^4\left(m-8\right)\left(m-4\right)$ .

Explanation of Solution

Given information:

A polynomial is given as

  $2m^6-24m^5+64m^4$

Calculation:

Consider the given polynomial.

  $2m^6-24m^5+64m^4$

Take out $2m^4$ common for each term.

  $2m^6-24m^5+64m^4=2m^4\left(m^2-12m+32\right)$

Now, break the coefficient of middle term $-12$ as sum or difference of two numbers such that product of the numbers is equal to the product of first term and constant term that is product of $1$ and $32$ .

Such numbers will be $-8$ and $-4$ .

Now, factorize $m^2-12m+32$ as shown:

  $\begin{array}{c}{m}^2-12m+32=m^2-4m-8m+32\\ =m\left(m-4\right)-8\left(m-4\right)\\ =\left(m-8\right)\left(m-4\right)\end{array}$

Therefore, $2m^6-24m^5+64m^4=2m^4\left(m-8\right)\left(m-4\right)$