Chapter 4.4, Problem 60E

To determine

To factor: the polynomial completely.

Answer to Problem 60E

The factorized form is $2p^8-12p^5+16p^2=2p^2\left(p^3-4\right)\left(p^3-2\right)$ .

Explanation of Solution

Given information:

A polynomial is given as

  $2p^8-12p^5+16p^2$

Calculation:

Consider the given polynomial.

  $2p^8-12p^5+16p^2$

Take out $2p^2$ common for each term.

  $2p^8-12p^5+16p^2=2p^2\left(p^6-6p^3+8\right)$

Now, break the coefficient of middle term $-6$ 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 $8$ .

Such numbers will be $-4$ and $-2$ .

Now, factorize

  $\left(p^6-6p^3+8\right)$ as shown:

  $\begin{array}{c}{p}^6-6p^3+8=p^6-4p^3-2p^3+8\\ =p^3\left(p^3-4\right)-2\left(p^3-4\right)\\ =\left(p^3-4\right)\left(p^3-2\right)\end{array}$

Therefore, $2p^8-12p^5+16p^2=2p^2\left(p^3-4\right)\left(p^3-2\right)$