Chapter 8.7, Problem 45PPS

a.

To determine

To factor the given polynomial

Answer to Problem 45PPS

  $h(h-6)(h-6)$

Explanation of Solution

Given:

Polynomial: $h^3-12h^2+36h$

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,

  $\Rightarrow h^3-12h^2+36h$

Taking out the common term h,

  $\Rightarrow h(h^2-12h+36)$

Substituting in the above formula for $a=1,b=-12,c=36$

  $\begin{array}{l}h=\frac{-(-12)\pm \sqrt{{(-12)}^2-4(1)(36)}}{2(1)}\\ h=\frac{12\pm \sqrt{144-144}}{2}\\ h=\frac{12}{2}=6\end{array}$

So, the equation can be written as,

  $\begin{array}{l}\Rightarrow h(h-6)(h-6)\\ $ \Rightarrow h{(h-6)}^2$\end{array}$

Conclusion:

Therefore, the polynomial is factored as $h(h-6)(h-6).$

b.

To determine

To explain the reason for the value of h to be greater than 6 inches.

Answer to Problem 45PPS

Yes, the value of h has to be greater than 6.

Explanation of Solution

Given:

From subpart (a),

The polynomial can be factored as $h(h-6)(h-6).$

Calculation / Explanation:

Consider the factored expression,

  $\Rightarrow h(h-6)(h-6)$

When the value of h is substituted as 6, the value of h becomes zero.

  $\begin{array}{l}\Rightarrow h(6-6)(6-6)\\ \Rightarrow 0\end{array}$

Therefore, the value of h has to be greater than 6 inches.

Conclusion:

Thus, the value of h has to be greater than 6.