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I see you've uploaded a photo of a math exercise sheet in Greek. Let's go through these exercises step by step. Here’s a breakdown and solution approach for each problem, starting with the first few. After each section, I'll confirm with you if you'd like more details.

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### **6. Να κάνετε τις πράξεις:**
This question involves simplifying polynomial expressions.

#### (α) \( 5x^2 - 2x^3 - 3x - 5x^2 + 2x + x^3 - 1 \)

1. Group like terms: 
   \[ (5x^2 - 5x^2) + (-2x^3 + x^3) + (-3x + 2x) + (-1) \]
2. Simplify:
   \[ 0x^2 + (-x^3) + (-x) - 1 \]
   \[ -x^3 - x - 1 \]

Final expression: \( -x^3 - x - 1 \)

#### (β) \( 2\alpha^2 - \beta^2 - 5\alpha\beta - 3\alpha^2 + \alpha\beta + 5\beta^2 \)

1. Group like terms: 
   \[ (2\alpha^2 - 3\alpha^2) + (-\beta^2 + 5\beta^2) + (-5\alpha\beta + \alpha\beta) \]
2. Simplify:
   \[ -\alpha^2 + 4\beta^2 - 4\alpha\beta \]

Final expression: \( -\alpha^2 + 4\beta^2 - 4\alpha\beta \)

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### **7. Να κάνετε τις πράξεις και μετά να γράψετε τα πολυώνυμα κατά τις φθίνουσες δυνάμεις:**
This question asks for simplification and arranging polynomials in descending powers.

#### (α) \( x^2 + (3x - x^3) - (3x - 2x^3) \)

1. Distribute the negative sign: 
   \[ x^2 + 3x - x^3 - 3x + 2x^3 \]
2. Group like terms: 
   \[ (-x^3 + 2x^3) + (3x - 3x) + x^2 \]
3. Simplify:
   \[ x^3 + x^2 \]

Final expression: \( x^3 + x^2 \)

#### (β) \( 3x^2 + 2 - (-x^3 - x^2 + 1) \)

1. Distribute the negative sign:
   \[ 3x^2 + 2 + x^3 + x^2 - 1 \]
2. Group like terms: 
   \[ (x^3) + (3x^2 + x^2) + (2 - 1) \]
3. Simplify:
   \[ x^3 + 4x^2 + 1 \]

Final expression: \( x^3 + 4x^2 + 1 \)

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### **8. Να κάνετε τις πράξεις:**
This exercise involves simplifying more complex polynomial expressions.

#### (α) \( \alpha \beta^2 - \left( \alpha^2 \beta - 2\alpha \beta \right) + (-2\alpha \beta^2 + \alpha \beta) - 3\alpha \beta^2 \)

1. Distribute the negative signs and combine terms:
   \[
   \alpha \beta^2 - \alpha^2 \beta + 2\alpha \beta - 2\alpha \beta^2 + \alpha \beta - 3\alpha \beta^2
   \]
2. Group like terms:
   \[
   \left( \alpha \beta^2 - 2\alpha \beta^2 - 3\alpha \beta^2 \right) + \left( -\alpha^2 \beta \right) + \left( 2\alpha \beta + \alpha \beta \right)
   \]
3. Simplify:
   \[
   -4\alpha \beta^2 - \alpha^2 \beta + 3\alpha \beta
   \]

Final expression: \( -4\alpha \beta^2 - \alpha^2 \beta + 3\alpha \beta \)

#### (β) \( \lambda^2 x - \lambda x^3 - \lambda x^3 - \lambda^2 x + 1 \)

1. Group like terms: 
   \[
   \left( \lambda^2 x - \lambda^2 x \right) + \left( -\lambda x^3 - \lambda x^3 \right) + 1
   \]
2. Simplify:
   \[
   -2\lambda x^3 + 1
   \]

Final expression: \( -2\lambda x^3 + 1 \)

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Would you like to continue with the next few exercises in the same manner? Let me know if you need clarification on any of the steps, and feel free to ask for further details.

Here are some relative questions for further exploration:

1. How do you simplify polynomials when variables and coefficients increase in complexity?
2. What is the general process for arranging terms in descending powers in polynomial expressions?
3. How do negative signs affect terms when distributing across parentheses?
4. Can you explain the importance of grouping like terms when simplifying expressions?
5. How do higher powers of variables (e.g., cubes and squares) impact simplification?

**Tip**: Always check for common factors before grouping terms—it can make simplification much easier!

I want help with all the exercises in the image.

Explore a step-by-step breakdown of polynomial simplification, combining like terms, and performing operations on expressions with multiple variables. The exercises cover important algebraic concepts, suitable for Grades 9-12.

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