Question
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression factors to:
\[
\frac{1}{2} d^{2} (d - 2 e^{2})(d^{2} + 2 d e^{2} + 4 e^{4})
\]
Solution
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:**
\[
\frac{1}{2} d^{2} \left( d - 2 e^{2} \right) \left( d^{2} + 2 d e^{2} + 4 e^{4} \right)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
Did you know that polynomials like the one you've presented can represent various physical phenomena? For instance, they might describe the motion of an object under the influence of forces in physics, where each term could represent different forces acting in a system. When simplifying or factoring such expressions, a common mistake is to overlook common factors, especially when the polynomial is multi-variable. Always check if there’s a greatest common factor first, as it can make the problem much simpler! In this case, for \( \frac{1}{2} d^{5} - 4 d^{2} e^{6} \), you could factor out \( \frac{1}{2} d^{2} \) to streamline your calculations.
