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Addition of Rational Expressions

Which rationa...

Question

Which rational expression should be subtracted from $\frac{4 x^{3} - 7 x^{2} + 5}{2 x - 1}$ to get $2 x^{2} - 5 x + 1$ ?

A

\frac{5 x^{2} - 7 x + 6}{2 x - 1}

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B

\frac{5 x^{2} + 7 x + 6}{2 x - 1}

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C

\frac{5 x^{2} - 7 x + 6}{2 x + 1}

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D

None of these

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Solution

The correct option is A $\frac{5 x^{2} - 7 x + 6}{2 x - 1}$
Let the rational expression be $A$ that must be subtracted from $\frac{4 x^{3} - 7 x^{2} + 5}{2 x - 1}$ to get $2 x^{2} - 5 x + 1$ , therefore, we have:

$\frac{4 x^{3} - 7 x^{2} + 5}{2 x - 1} - A = 2 x^{2} - 5 x + 1 \Rightarrow A = \frac{4 x^{3} - 7 x^{2} + 5}{2 x - 1} - \frac{2 x^{2} - 5 x + 1}{1} \Rightarrow A = \frac{(4 x^{3} - 7 x^{2} + 5) - (2 x - 1) (2 x^{2} - 5 x + 1)}{2 x - 1} (T a k i n g L C M) \Rightarrow A = \frac{(4 x^{3} - 7 x^{2} + 5) - (4 x^{3} - 10 x^{2} + 2 x - 2 x^{2} + 5 x - 1)}{2 x - 1}$
$\Rightarrow A = \frac{(4 x^{3} - 7 x^{2} + 5) - (4 x^{3} - 12 x^{2} + 7 x - 1)}{2 x - 1} \Rightarrow A = \frac{4 x^{3} - 7 x^{2} + 5 - 4 x^{3} + 12 x^{2} - 7 x + 1}{2 x - 1} \Rightarrow A = \frac{5 x^{2} - 7 x + 6}{2 x - 1}$

Hence, $\frac{5 x^{2} - 7 x + 6}{2 x - 1}$ should be subtracted from $\frac{4 x^{3} - 7 x^{2} + 5}{2 x - 1}$ to get $2 x^{2} - 5 x + 1$ .

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