Question 1209748: Factor 3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Factoring that expression directly is quite difficult. It's a complex combination of terms, and there's no immediately obvious grouping or pattern that leads to a clean factorization. It's likely that this expression, as given, does *not* have a simple factorization into polynomials with integer or even rational coefficients.
Here's why and what approaches you could consider:
1. **Complexity:** The mix of x, y, x², y², xy, x²y, and y³ terms makes it very challenging to find a factorization. There's no standard factoring technique that readily applies.
2. **No Obvious Grouping:** While some terms might seem like they could be grouped, it doesn't lead to a common factor that can be pulled out.
3. **Rational Root Theorem (for single-variable polynomials):** This theorem helps find rational roots of polynomials with integer coefficients. However, it applies to polynomials in *one* variable. Our expression has two variables, x and y, so this theorem isn't directly applicable.
4. **Numerical/Computational Approach:** If you have specific values for x and y in mind, substituting them might simplify the expression. However, this doesn't lead to a general factorization. Computer algebra systems (CAS) like Wolfram Alpha, Mathematica, or Maple are designed to handle such complex expressions. They might be able to find a factorization if one exists, or at least simplify the expression.
5. **Possible Error:** It's also worth double-checking if the expression is written correctly. A small typo can drastically change the factorability of an expression.
**In summary:** Unless there's a trick or a specific substitution that simplifies the expression (which I don't see), it's highly unlikely that this expression has a simple factorization. Using a CAS is the most practical approach if you need to work with this expression further.
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