Chapter 6.5, Problem 1BCYP

To determine

To find: the factor of the given polynomial and check whether the polynomial is factored or prime.

Answer to Problem 1BCYP

  $-54w^4-250w{z}^3=-27w\left(w+5z\right)\left(w^2-5wz+25z^2\right)$ .

Explanation of Solution

Given:

The given polynomial is $-54w^4-250w{z}^3$ .

Calculation:

The given polynomial is $-54w^4-250w{z}^3$ .

Now, to find the factor of the given polynomial. Therefore,

  $\begin{array}{l}-54w^4-250w{z}^3\left\{\text{given polynomial}\right\}\\ =-27w\left(w^3+125z^3\right)\left\{\text{factor out the GCF}\right\}\end{array}$

Since are both perfect cubes, so write the factor of the sum of two cubes.

  $\begin{array}{l}{w}^3+125z^3\\ ={\left(w\right)}^3-{\left(5z\right)}^3\{\because w^3={\left(w\right)}^3;125z^3={\left(5z\right)}^3\}\\ \left\{a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)\right\}\\ =\left(w+5z\right)\left(w^2-w\left(5z\right)+{\left(5z\right)}^2\right)\\ =\left(w+5z\right)\left(w^2-5wz+25z^2\right)\left\{\text{simplified form}\right\}\end{array}$

Thus, $-54w^4-250w{z}^3=-27w\left(w+5z\right)\left(w^2-5wz+25z^2\right)$

Hence, the factor of the given polynomial $-54w^4-250w{z}^3=-27w\left(w+5z\right)\left(w^2-5wz+25z^2\right)$ .