Chapter 6.8, Problem 4STP

To determine

To calculate: The simplified form of the equation: $x^3-37x-84$ .

Answer to Problem 4STP

The simplified functions of $x^3-37x-84$ is $x=-3,x=-4,x=7$ .

Explanation of Solution

Given information:

The simplified form of the equation: $x^3-37x-84$ ..”

In the given options which is Not the solution of the cubic equation below:

  $\begin{array}{l}F-4H6\\ G-3j7\end{array}$

Formula used:

When two or more functions are in the form of equation, apply the expanded form of the monomials.

Calculation:

Consider the provided equation, $x^3-37x-84$

So, the given expression can be simplified as,

  $x^3-37x-84$

Observe to solve the expression by expend monomials into a polynomials and add and subtract $3$ in the equation.

  $\begin{array}{l}{x}^3-37x-84\\ x^3+3x^2-3x^2-9x-28x-84\end{array}$

Now group to the values:

  $\begin{array}{l}{x}^3+3x^2-3x^2-9x-28x-84\\ \left(x^3+3x^2\right)-\left(3x^2-9x\right)-\left(28x-84\right)\end{array}$

Factor out the greatest common factor:

  $\begin{array}{l}\left(x^3+3x^2\right)-\left(3x^2-9x\right)-\left(28x-84\right)\\ x^2\left(x+3\right)-3x\left(x+3\right)-28\left(x+3\right)\end{array}$

Here the equation becomes:

  $\begin{array}{l}{x}^2\left(x+3\right)-3x\left(x+3\right)-28\left(x+3\right)\\ \left(x+3\right)\left(x^2-3x-28\right)\end{array}$

Now take $\left(x+3\right)=A$ , and $\left(x^2-3x-28\right)=B$ .

Now solve the part $B$ separately by the middle term factorization method:

  $\begin{array}{l}{x}^2-3x-28\\ x^2-7x+4x-28\end{array}$

Factor out the greatest common factor:

  $\begin{array}{l}{x}^2-7x+4x-28\\ x\left(x-7\right)+4\left(x-7\right)\\ \left(x-7\right)\left(x+4\right)\end{array}$

Now the equation $\left(x+3\right)\left(x^2-3x-28\right)$ becomes:

  $\begin{array}{l}\left(x+3\right)\left(x^2-3x-28\right)\\ \left(x+3\right)\left(x-7\right)\left(x+4\right)\end{array}$

Now either:

  $\begin{array}{c}\left(x+3\right)=0\\ x=-3\\ \left(x-7\right)=0\\ x=7\end{array}$

Or,

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

Since these are the solutions of the function:

  $x=-3,x=-4,x=7$

Hence, the simplified functions of $x^3-37x-84$ is $x=-3,x=-4,x=7$ .