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Add the following fractions: (u2 - v2)/(uv) + (v - 3uv - u2)/(v2)

Mathematics

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Answer

−u3−2u2v+uv−v3uv2\frac{-u^3 - 2u^2v + uv - v^3}{uv^2}

Step-by-step solution

5 steps

  1. Find a common denominator for the two fractions.

    The denominators are uvuv and v2v^2. The least common multiple (LCM) of uvuv and v2v^2 is uv2uv^2.

  2. Rewrite each fraction with the common denominator.

    For the first fraction, multiply the numerator and denominator by vv:

    u2−v2uv=(u2−v2)⋅vuv⋅v=u2v−v3uv2\frac{u^2 - v^2}{uv} = \frac{(u^2 - v^2) \cdot v}{uv \cdot v} = \frac{u^2v - v^3}{uv^2}

    For the second fraction, multiply the numerator and denominator by uu:

    v−3uv−u2v2=(v−3uv−u2)⋅uv2⋅u=uv−3u2v−u3uv2\frac{v - 3uv - u^2}{v^2} = \frac{(v - 3uv - u^2) \cdot u}{v^2 \cdot u} = \frac{uv - 3u^2v - u^3}{uv^2}

  3. Add the rewritten fractions.

    u2v−v3uv2+uv−3u2v−u3uv2=u2v−v3+uv−3u2v−u3uv2\frac{u^2v - v^3}{uv^2} + \frac{uv - 3u^2v - u^3}{uv^2} = \frac{u^2v - v^3 + uv - 3u^2v - u^3}{uv^2}

  4. Combine like terms in the numerator.

    Combine the terms involving u2vu^2v: u2v−3u2v=−2u2vu^2v - 3u^2v = -2u^2v.
    The numerator becomes: −u3−2u2v+uv−v3-u^3 - 2u^2v + uv - v^3.

  5. Write the simplified expression.

    −u3−2u2v+uv−v3uv2\frac{-u^3 - 2u^2v + uv - v^3}{uv^2}
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