Chapter 6.5, Problem 5CYU

To determine

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

Answer to Problem 5CYU

  $12q{w}^3-12q^4=12q\left(w-q\right)\left(w^2+wq+q^2\right)$ .

Explanation of Solution

Given:

The given polynomial is $12q{w}^3-12q^4$ .

Calculation:

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

  $\begin{array}{l}12q{w}^3-12q^4\left\{\text{given polynomial}\right\}\\ =12q\left(w^3-q^3\right)\left\{\text{factor out the GCF}\right\}\end{array}$

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

  $\begin{array}{l}{w}^3-q^3\\ ={\left(w\right)}^3+{\left(q\right)}^3\{\because 8g^3={\left(2g\right)}^3;h^3={\left(h\right)}^3\}\\ \left\{a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)\right\}\\ =\left(w-q\right)\left(w^2+wq+q^2\right)\left\{\text{simplified form}\right\}\end{array}$

Thus, $12q{w}^3-12q^4=12q\left(w-q\right)\left(w^2+wq+q^2\right)$

Hence, the factor of the given polynomial $12q{w}^3-12q^4=12q\left(w-q\right)\left(w^2+wq+q^2\right)$ .