Chapter A.2, Problem 82E

Solution

To factor completely $z-8z^4$ .

  $-z\left(\left(2z-1\right)\left(4z^2+2z+1\right)\right)$

Given:

The expression, $z-8z^4$

Concept Used:

• Like terms are the terms that contain the same variable with similar exponents on it. Constants are like terms as well.
• In any expression, terms are separated by plus or minus signs.
• Coefficients are the constants in front of the variable terms.

Calculation:

In order to find the factor completely, first rearrange the polynomial and then out the common factor and then solving the equatiom by using the formula $a^3-b^3=\left(a-b\right)\left(a^2+b^2+ab\right)$ as shown below:

  $\begin{array}{l}z-8z^4\\ -8z^4+z\\ -z\left(8z^3-1\right)\\ -z\left({\left(2z\right)}^3-1^3\right)\\ -z\left(\left(2z-1\right)\left({\left(2z\right)}^2+1^2+2z\cdot 1\right)\right)\\ -z\left(\left(2z-1\right)\left(4z^2+1+2z\right)\right)\end{array}$

The original expression now become

  $-z\left(\left(2z-1\right)\left(4z^2+2z+1\right)\right)$