4. $$ \frac{1}{2} d^{5}-4 d^{2} e^{6} $$

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UpStudy AI · Accepted Answer

Certainly! Let's factor the expression step by step.

**Given Expression:**
$$
\frac{1}{2} d^{5} - 4 d^{2} e^{6}
$$

**Step 1: Factor Out the Greatest Common Factor (GCF)**

First, identify the GCF of the two terms.

- **Numerical Coefficients:** The GCF of $$\frac{1}{2}$$ and $$4$$ is $$\frac{1}{2}$$.
- **Variables:**
  - For $$d$$: The lowest power of $$d$$ is $$d^{2}$$.
  - For $$e$$: The term $$e^{6}$$ is only present in the second term, so it doesn't contribute to the GCF.

**Factor out $$\frac{1}{2} d^{2}$$:**
$$
\frac{1}{2} d^{5} - 4 d^{2} e^{6} = \frac{1}{2} d^{2} \left( d^{3} - 8 e^{6} \right)
$$

**Step 2: Factor the Remaining Polynomial**

Now, focus on factoring $$d^{3} - 8 e^{6}$$.

Notice that $$8 e^{6}$$ can be written as $$(2 e^{2})^{3}$$, making the expression a **difference of cubes**:
$$
d^{3} - (2 e^{2})^{3}
$$

The difference of cubes can be factored using the formula:
$$
a^{3} - b^{3} = (a - b)(a^{2} + a b + b^{2})
$$

Applying this formula:
$$
d^{3} - (2 e^{2})^{3} = (d - 2 e^{2})(d^{2} + 2 d e^{2} + 4 e^{4})
$$

**Step 3: Combine the Factors**

Substitute the factored form back into the expression:
$$
\frac{1}{2} d^{2} (d - 2 e^{2})(d^{2} + 2 d e^{2} + 4 e^{4})
$$

**Final Factored Form:**
$$
\frac{1}{2} d^{2} \left( d - 2 e^{2} \right) \left( d^{2} + 2 d e^{2} + 4 e^{4} \right)
$$
The expression factors to: $$ \frac{1}{2} d^{2} (d - 2 e^{2})(d^{2} + 2 d e^{2} + 4 e^{4}) $$

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