Question

Determine Whether the statement is true or false. If it is true explain

why. If it is false explain why or give an example that disproves the statement.

1. \(\frac{{x\left( {{x^2} + 4} \right)}}{{{x^2} - 4}}\)can be put in the form of \(\frac{A}{{x + 2}} + \frac{B}{{x - 2}}\)

Short answer

False, the value \(\frac{{x\left( {{x^2} + 4} \right)}}{{{x^2} - 4}}\) cannot be put in the form of \(\frac{A}{{x + 2}} + \frac{B}{{x - 2}}\)

Step-by-step solution

Given parameters

We need to check if the equation \(\frac{{x\left( {{x^2} + 4} \right)}}{{{x^2} - 4}}\) can be written in the form of \(\frac{A}{{x + 2}} + \frac{B}{{x - 2}}\)

Proving that \(\frac{{x\left( {{x^2} + 4} \right)}}{{{x^2} - 4}}\) cannot be written in the form of \(\frac{A}{{x + 2}} + \frac{B}{{x - 2}}\)

Rewrite the given equation as \(\frac{{{x^3} + 4x}}{{{x^2} - 4}}\) --- Equation 1

By applying long division for the equation

\({x^2} - 4\mathop{\left){\vphantom{1\begin{array}{l}{x^3} + 4x\\{x^3} - 4x\\\_\_\_\_\_\_\\8x\\\_\_\_\_\_\_\end{array}}}\right.

\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{array}{l}{x^3} + 4x\\{x^3} - 4x\\\_\_\_\_\_\_\\8x\\\_\_\_\_\_\_\end{array}}}}

\limits^{\displaystyle\,\,\, x}\)

From the division we can rewrite the Equation 1 as \(\frac{{{x^3} + 4x}}{{{x^2} - 4}} = x + \frac{{8x}}{{{x^2} - 4}}\)

The value \(\frac{{8x}}{{{x^2} - 4}}\) can be written in terms of partial equation as \(\frac{A}{{x - 2}} + \frac{B}{{x + 2}}\)

The resultant value cannot be written in terms of \(\frac{A}{{x + 2}} + \frac{B}{{x - 2}}\)

Hence, \(\frac{{x\left( {{x^2} + 4} \right)}}{{{x^2} - 4}}\) can be written in the form of \(\frac{A}{{x + 2}} + \frac{B}{{x - 2}}\).