Chapter 8.7, Problem 38PPS

To determine

To factor the given polynomial

Answer to Problem 38PPS

  $3k(k-4)(k-4)$

Explanation of Solution

Given:

Polynomial: $3k^3-24k^2+48k$

Formula used:

Roots of the polynomial of the form $a{x}^2+bx+c=a(x-x_1)(x-x_2)$ are

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

Consider the polynomial,

  $\Rightarrow 3k^3-24k^2+48k$

Find the common term in each term,

  $\Rightarrow 3k(k^2-8k+16)$

Substituting in the above formula, where $a=1,b=-8,c=16$

  $\begin{array}{l}k=\frac{-(-8)\pm \sqrt{{(-8)}^2-4(1)(16)}}{2(1)}\\ k=\frac{8\pm \sqrt{64-64)}}{2}\\ k=\frac{8}{2}=4\end{array}$

The roots are

  $k_1=k_2=4$

Thus, the factors of the equation are $k^2-8k+16=(k-4)(k-4).$

Conclusion:

Therefore, the factors of the given equation are $3k(k-4)(k-4).$