Chapter 2.3, Problem 70E

a.

To determine

Verify the factors of $f\left(x\right)$ .

Answer to Problem 70E

  $Yes,theyarethefactorsofthepolynomial$

Explanation of Solution

Calculation: Here, we will consider the following function and factors of $f\left(x\right)$ :

  $f\left(x\right)=x^3+3x^2-48x-144$ And $\left(x+4\sqrt{3}\right),\left(x+3\right)$

Now, we will divide $x^3+3x^2-48x-144$ by $\left(x+4\sqrt{3}\right)$ using synthetic division method:

  $\begin{array}{lllll}\hfill & 1\hfill & 3\hfill & -48\hfill & -144\hfill \\ 4\sqrt{3}\hfill & \hfill & -4\sqrt{3}\hfill & 48-12\sqrt{3}\hfill & 144\hfill \\ \hfill & 1\hfill & 3-4\sqrt{3}\hfill & -12\sqrt{3}\hfill & 0\hfill \end{array}$

Hence, the remainder is zero so $\left(x+4\sqrt{3}\right)$ is a factor of the polynomial $x^3+3x^2-48x-144$ .

Now, we will again divide $x^3+3x^2-48x-144$ by $\left(x+3\right)$ using same division method:

  $\begin{array}{lllll}\hfill & 1\hfill & 3\hfill & -48\hfill & -144\hfill \\ -3\hfill & \hfill & -3\hfill & 0\hfill & 144\hfill \\ \hfill & 1\hfill & 0\hfill & -48\hfill & 0\hfill \end{array}$

Hence, again the remainder is zero so $\left(x+3\right)$ is a factor of the polynomial $x^3+3x^2-48x-144$ .

b.

To determine

Find the remaining factor(s) of $f\left(x\right)$ .

Answer to Problem 70E

  $\left(x-4\sqrt{3}\right)$

Explanation of Solution

Calculation: Here, from the above synthetic division we will rewrite the polynomial $x^3+3x^2-48x-144$ as:

  $x^3+3x^2-48x-144=\left(x+3\right)\left(x^2-48\right)$

  $=\left(x+3\right)\left(x^2-{\left(4\sqrt{3}\right)}^2\right)$

  $=\left(x+3\right)\left(x-4\sqrt{3}\right)\left(x+4\sqrt{3}\right)$

Hence, the remaining factor is $\left(x-4\sqrt{3}\right)$ .

c.

To determine

Write the complete factorization of $f\left(x\right)$ .

Answer to Problem 70E

  $\left(x+3\right)\left(x-4\sqrt{3}\right)\left(x+4\sqrt{3}\right)$

Explanation of Solution

Calculation:

As, $\left(x+3\right),\left(x-4\sqrt{3}\right)$ and $\left(x+4\sqrt{3}\right)$ are the factors of the function $f\left(x\right)$ .

Thus, the complete factorization of the function $f$ will be:

  $f\left(x\right)=\left(x+3\right)\left(x-4\sqrt{3}\right)\left(x+4\sqrt{3}\right)$

Hence, the complete factorization of the function $f$ is $\left(x+3\right)\left(x-4\sqrt{3}\right)\left(x+4\sqrt{3}\right)$ .

d.

To determine

List all real zeros of $f$ .

Answer to Problem 70E

  $-3,4\sqrt{3}and-4\sqrt{3}$

Explanation of Solution

Calculation:

Here, to find the zeroes of $f$ , we will put $f\left(x\right)=0$ and solve for $x$ :

  $f\left(x\right)=0$

  $\left(x+3\right)\left(x-4\sqrt{3}\right)\left(x+4\sqrt{3}\right)=0$

Now, we will set the first factor to zero and solve for $x$ :

  $\begin{array}{l}\left(x+3\right)=0\\ x=-3\end{array}$

Now, we will set the second factor to zero and solve for $x$ :

  $\begin{array}{l}\left(x-4\sqrt{3}\right)=0\\ x=4\sqrt{3}\end{array}$

Now, we will set the third factor to zero and solve for $x$ :

  $\begin{array}{l}\left(x+4\sqrt{3}\right)=0\\ x=-4\sqrt{3}\end{array}$

Hence, the zeroes of the function are $-3,4\sqrt{3}and-4\sqrt{3}$ .

e.

To determine

Confirm your results by using a graphing utility.

Answer to Problem 70E

  

Explanation of Solution

Calculation:

Here, the graph of the function $f\left(x\right)=x^3+3x^2-48x-144$ is:

  

Now, from the above graph we have observed that the graph of the function intersects $x-axis$ at three points:

  $-3,4\sqrt{3}and-4\sqrt{3}$

Hence, the results are verified.