\(\frac{k(k+3)}{4(k+1)(k+2)}+\frac{1}{k(k+1)(k+2)}\)

asked by guest
on Nov 26, 2024 at 6:21 pm



You asked:

Evaluate the expression: \(\frac{k \cdot \left(k + 3\right)}{4 \cdot \left(k + 1\right) \cdot \left(k + 2\right)} + \frac{1}{k \cdot \left(k + 1\right) \cdot \left(k + 2\right)}\)

MathBot Answer:

Evaluated



\(\displaystyle \frac{k \cdot \left(k + 3\right)}{4 \cdot \left(k + 1\right) \cdot \left(k + 2\right)} + \frac{1}{k \cdot \left(k + 1\right) \cdot \left(k + 2\right)} = \frac{k \left(k + 3\right)}{4 \left(k + 1\right) \left(k + 2\right)} + \frac{1}{k \left(k + 1\right) \left(k + 2\right)} \)

Expanded

\[\frac{k \cdot \left(k + 3\right)}{4 \cdot \left(k + 1\right) \cdot \left(k + 2\right)} + \frac{1}{k \cdot \left(k + 1\right) \cdot \left(k + 2\right)} = \frac{k^{2}}{4 k^{2} + 12 k + 8} + \frac{3 k}{4 k^{2} + 12 k + 8} + \frac{1}{k^{3} + 3 k^{2} + 2 k}\]

Factored

\[\frac{k \cdot \left(k + 3\right)}{4 \cdot \left(k + 1\right) \cdot \left(k + 2\right)} + \frac{1}{k \cdot \left(k + 1\right) \cdot \left(k + 2\right)} = \frac{k^{3} + 3 k^{2} + 4}{4 k \left(k + 1\right) \left(k + 2\right)}\]

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