Determine whether the equations are equivalent. 
  1. 2𝑥2+3𝑥3=5𝑥5 

  2. 4ℎ2+3ℎ2=7ℎ2 

  3. 2𝐴𝐵2+3𝐴2𝐵=5𝐴2𝐵2 

  4. 3𝑏+2𝑏2=5𝑏3 

To determine whether the equations are equivalent, we need to simplify each equation and check if they hold true under certain conditions. 

1. **Equation 1: \(2x^2 + 3x^3 = 5x^5\)**
   - Rearranging gives: \(2x^2 + 3x^3 - 5x^5 = 0\)
   - Factoring out \(x^2\): \(x^2(2 + 3x - 5x^3) = 0\)
   - This means \(x^2 = 0\) (which gives \(x = 0\)) or \(2 + 3x - 5x^3 = 0\).

2. **Equation 2: \(4h^2 + 3h^2 = 7h^2\)**
   - Simplifying gives: \(7h^2 = 7h^2\)
   - This is always true for any \(h\).

3. **Equation 3: \(2AB^2 + 3A^2B = 5A^2B^2\)**
   - Rearranging gives: \(2AB^2 + 3A^2B - 5A^2B^2 = 0\)
   - Factoring out \(AB\): \(AB(2B + 3A - 5AB) = 0\)
   - This means \(AB = 0\) (which gives \(A = 0\) or \(B = 0\)) or \(2B + 3A - 5AB = 0\).

4. **Equation 4: \(3b + 2b^2 = 5b^3\)**
   - Rearranging gives: \(3b + 2b^2 - 5b^3 = 0\)
   - Factoring out \(b\): \(b(3 + 2b - 5b^2) = 0\)
   - This means \(b = 0\) or \(3 + 2b - 5b^2 = 0\).

### Summary of Equivalence:
- **Equation 1:** Not equivalent to others; depends on polynomial roots.
- **Equation 2:** Always holds true; it’s equivalent to itself.
- **Equation 3:** Not equivalent to others; depends on polynomial roots.
- **Equation 4:** Not equivalent to others; depends on polynomial roots.

### Conclusion:
Only Equation 2 is universally true. The others depend on the specific values of the variables.

Would you like more details on any specific equation? Here are some questions to consider:
1. What are the roots of the polynomials in Equations 1, 3, and 4?
2. How can we graphically represent these equations to visualize their relationships?
3. What conditions make Equations 1, 3, and 4 true?
4. Can you find specific values for variables that satisfy all equations?
5. How do these equations relate to systems of equations in higher dimensions?

**Tip:** Always check for common factors in polynomial equations to simplify and analyze their relationships more effectively.

Determine whether the equations are equivalent. 1. 2𝑥^2 + 3𝑥^3 = 5𝑥^5, 2. 4ℎ^2 + 3ℎ^2 = 7ℎ^2, 3. 2𝐴𝐵^2 + 3𝐴^2𝐵 = 5𝐴^2𝐵^2, 4. 3𝑏 + 2𝑏^2 = 5𝑏^3

Analyze and determine if the given polynomial equations are equivalent by simplifying and factoring each. Explore polynomial expressions and the Zero Product Property, suitable for students in Grades 9-10.

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