Question

subtract these two fractions and simplify the answer.

2x+3/x^4-x+2/x^3

Answer 1

To subtract the fractions \(\frac{2x+3}{x^4}\) and \(\frac{x-2}{x^3}\), we need a common denominator. The common denominator will be \(x^4\) since it is the highest power of \(x\) in the denominators.

Let's rewrite the second fraction with a common denominator of \(x^4\):

\[
\frac{x-2}{x^3} = \frac{(x-2) \cdot x}{x^4} = \frac{x^2 - 2x}{x^4}
\]

Now we can subtract the two fractions:

\[
\frac{2x+3}{x^4} - \frac{x^2 - 2x}{x^4} = \frac{(2x+3) - (x^2 - 2x)}{x^4}
\]

Distributing the negative sign in the numerator:

\[
\frac{2x + 3 - x^2 + 2x}{x^4} = \frac{-x^2 + 4x + 3}{x^4}
\]

Rearranging the numerator:

\[
\frac{-x^2 + 4x + 3}{x^4} = \frac{-(x^2 - 4x - 3)}{x^4}
\]

Next, we can factor the quadratic in the numerator:

To factor \(x^2 - 4x - 3\), we need two numbers that multiply to \(-3\) and add to \(-4\). The numbers \(-5\) and \(1\) fit this criterion:

\[
x^2 - 4x - 3 = (x - 5)(x + 1)
\]

Thus, our expression becomes:

\[
\frac{-(x - 5)(x + 1)}{x^4}
\]

Putting that all together, we have our final simplified form:

\[
-\frac{(x - 5)(x + 1)}{x^4}
\]

So the result of the subtraction \(\frac{2x+3}{x^4} - \frac{x-2}{x^3}\) simplified is:

\[
-\frac{(x - 5)(x + 1)}{x^4}
\]