Chapter 8.7, Problem 56PFA

a.

To determine

To write the polynomial in factored form

Answer to Problem 56PFA

  $l(l-10)(l-6)$

Explanation of Solution

Given:

Polynomial: $l^3-16l^2+60l$

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,

  $\Rightarrow l^3-16l^2+60l$

Taking out the common terms,

  $\begin{array}{l}\Rightarrow l^3-16l^2+60l\\ \Rightarrow l(l^2-16l+60)\end{array}$

Substituting in the above formula for $a=1,b=-16,c=60$

  $\begin{array}{l}\Rightarrow l=\frac{-(-16)\pm \sqrt{{(-16)}^2-4(1)(60)}}{2(1)}\\ \Rightarrow l=\frac{16\pm \sqrt{256-240}}{2}\\ \Rightarrow l=\frac{16\pm \sqrt{16}}{2}\\ \Rightarrow l=\frac{16\pm 4}{2}\end{array}$

Hence, the roots are

  $\begin{array}{l}\Rightarrow l_1=\frac{16+4}{2}=\frac{20}{2}=10\\ \Rightarrow l_2=\frac{16-4}{2}=\frac{12}{2}=6\end{array}$

Therefore, the polynomial is factored as

  $\Rightarrow l^2-16l+60=(l-10)(l-6)$

Thus, the polynomial is completely factored as

  $\Rightarrow l^3-16l^2+60l=l(l-10)(l-6)$

Conclusion:

Therefore, the completely factored form is $l(l-10)(l-6).$

b.

To determine

To use factor to write the expression for width and height in terms of length.

Answer to Problem 56PFA

Height: ( l − 6 )

Width: ( l − 10 )

Explanation of Solution

Given:

From subpart (a),

  $l(l-10)(l-6)$

Calculation:

Let l be the length of the prism

Height is ( l − 6)

Width is ( l − 10)

Conclusion:

Therefore, the height and width is expressed in terms of the length.