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.

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

Short answer

The given statement is False.

Step-by-step solution

Irreducible quadratic factor

If\(Q\left( x \right)\) has the factor\(a{x^2} + bx + c\), where\({b^2} - 4ac < 0\), then in addition to the partial fractions in\(\frac{{R\left( x \right)}}{{Q\left( x \right)}} = \frac{{{A_1}}}{{{a_1}x + {b_1}}} + \frac{{{A_2}}}{{{a_2}x + {b_2}}} + \ldots \frac{{{A_k}}}{{{a_k}x + {b_k}}}\) and\(\frac{{{A_1}}}{{{a_1}x + {b_1}}} + \frac{{{A_2}}}{{{{\left( {{a_2}x + {b_2}} \right)}^2}}} + \ldots \frac{{{A_k}}}{{{{\left( {{a_k}x + {b_k}} \right)}^r}}}\), the expression for\(\frac{{R\left( x \right)}}{{Q\left( x \right)}}\) will have a term of the form\(\frac{{Ax + B}}{{a{x^2} + bx + c}}\).

Explanation

It is given that \(\frac{{{x^2} - 4}}{{x\left( {{x^2} + 4} \right)}}\)and we know that it can be written as in the form\(\frac{{{x^2} - 4}}{{x\left( {{x^2} + 4} \right)}} = \frac{A}{x} + \frac{{Bx + C}}{{{x^2} + 4}}\)

Since the denominator has a linear factor has \({x^2} + 4\)in the form of \(ax + b\), the partial fractions of \(\frac{{{x^2} - 4}}{{x\left( {{x^2} + 4} \right)}}\)can be represented in the form \(\frac{{{x^2} - 4}}{{x\left( {{x^2} + 4} \right)}} = \frac{A}{x} + \frac{{Bx + C}}{{{x^2} + 4}}\).

Thus, the statement is FALSE.