Chapter 6.8, Problem 1ACYP

To determine

To calculate: The simplified form of the expression, $g\left(x\right)=3x^3-4x+10$ .

Answer to Problem 1ACYP

The simplified form of the expression is $3x^3-4x+10$ .

Explanation of Solution

Given information:

The expression, $3x^3-4x+10$

Calculation:

Consider the expression, $g\left(x\right)=3x^3-4x+10$

Let’s factor the expression with the help of factorization:

  $\begin{array}{c}g\left(x\right)=3x^3-4x+10\\ =\left(3x^3-4x\right)+10\end{array}$

Here $f\left(x\right)=0$ .

Then,

  $3x^3-4x+10=0$

Accordance to the rational roots test. It states that if a polynomial zeros for a rational number 1 $p/q$ then $p$ is a factor of the trilling constant and $q$ is the factor of leading coefficient.

The leading coefficient is $3$ and trailing content is $10$ .

Then the factor of leading coefficient is: $1,3$ .

Then the Factor of trilling constant is: $1,2,5,10$ .

Here the last two terms subtracting values from the cube of the first and want to get zero,

So as have to take the positive number, and mostly take the less than mid-way.

So take a startup with $12$ and check the result, then again tried to get zero.

  $\begin{array}{c}3x^3-4x+10=0\\ 3{\left(12\right)}^3-4\left(12\right)+10=0\\ 3\left(1728\right)-\left(48\right)+10=0\\ 5184-48+10=0\end{array}$

To get zero:

  $\begin{array}{c}5184-48+10=0\\ 5146=0\end{array}$

So the value is not zero its too grater than zero.

So check with the value $2$ .

  $\begin{array}{c}3x^3-4x+10=0\\ 3{\left(2\right)}^3-4\left(2\right)+10=0\\ 3\left(8\right)-\left(8\right)+10=0\\ 24-8+10=0\end{array}$

To get zero:

  $\begin{array}{c}24-8+10=0\\ 26=0\end{array}$

So the value is not zero its too grater than zero.

So check with the value $-2$ .

  $\begin{array}{c}3x^3-4x+10=0\\ 3{\left(-2\right)}^3-4\left(-2\right)+10=0\\ 3\left(-8\right)+8+10=0\\ -24+8+10=0\end{array}$

To get zero:

  $\begin{array}{c}-24+8+10=0\\ -6=0\end{array}$

So the value is not zero its too grater than zero.

So check with the value $-1$ .

  $\begin{array}{c}3x^3-4x+10=0\\ 3{\left(-1\right)}^3-4\left(-1\right)+10=0\\ -3+4+10=0\end{array}$

To get zero:

  $\begin{array}{c}-3+4+10=0\\ 11=0\end{array}$

Here no polynomial roots are found.

There is no further simplification of : $3x^3-4x+10$.

Thus, the simplified form of the expression is: $3x^3-4x+10$