SOLUTION: Math worksheet addition and subtraction of polynomials ... - Free Printable

Problem: Simplify each polynomial expression by adding or subtracting like terms.

#### Step-by-Step Solutions:

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1. \( (5x^2 - 2x - 2) + (3x^2 + x - 1) \)

- Combine like terms:
- \( x^2 \)-terms: \( 5x^2 + 3x^2 = 8x^2 \)
- \( x \)-terms: \( -2x + x = -x \)
- Constant terms: \( -2 - 1 = -3 \)

- Simplified expression: \( 8x^2 - x - 3 \)

Answer: \( 8x^2 - x - 3 \)

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2. \( (2x^2 + 5x - 3) + (12x^2 - 4x + 4) \)

- Combine like terms:
- \( x^2 \)-terms: \( 2x^2 + 12x^2 = 14x^2 \)
- \( x \)-terms: \( 5x - 4x = x \)
- Constant terms: \( -3 + 4 = 1 \)

- Simplified expression: \( 14x^2 + x + 1 \)

Answer: \( 14x^2 + x + 1 \)

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3. \( (10x^3 - 6x^2 + 2x) + (5x^2 + 2x + 3) \)

- Combine like terms:
- \( x^3 \)-terms: \( 10x^3 \) (no other \( x^3 \)-terms)
- \( x^2 \)-terms: \( -6x^2 + 5x^2 = -x^2 \)
- \( x \)-terms: \( 2x + 2x = 4x \)
- Constant terms: \( 3 \) (no other constant terms)

- Simplified expression: \( 10x^3 - x^2 + 4x + 3 \)

Answer: \( 10x^3 - x^2 + 4x + 3 \)

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4. \( (7y^2 + 4y - 1) - (2y^2 + 4y - 12) \)

- Distribute the negative sign:
- \( 7y^2 + 4y - 1 - 2y^2 - 4y + 12 \)

- Combine like terms:
- \( y^2 \)-terms: \( 7y^2 - 2y^2 = 5y^2 \)
- \( y \)-terms: \( 4y - 4y = 0 \)
- Constant terms: \( -1 + 12 = 11 \)

- Simplified expression: \( 5y^2 + 11 \)

Answer: \( 5y^2 + 11 \)

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5. \( (3m^2 - 2m - 2) - (2m^2 + m + 1) \)

- Distribute the negative sign:
- \( 3m^2 - 2m - 2 - 2m^2 - m - 1 \)

- Combine like terms:
- \( m^2 \)-terms: \( 3m^2 - 2m^2 = m^2 \)
- \( m \)-terms: \( -2m - m = -3m \)
- Constant terms: \( -2 - 1 = -3 \)

- Simplified expression: \( m^2 - 3m - 3 \)

Answer: \( m^2 - 3m - 3 \)

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6. \( (5y^3 + 2y - 12) - (3y^2 - 5y - 7) \)

- Distribute the negative sign:
- \( 5y^3 + 2y - 12 - 3y^2 + 5y + 7 \)

- Combine like terms:
- \( y^3 \)-terms: \( 5y^3 \) (no other \( y^3 \)-terms)
- \( y^2 \)-terms: \( -3y^2 \) (no other \( y^2 \)-terms)
- \( y \)-terms: \( 2y + 5y = 7y \)
- Constant terms: \( -12 + 7 = -5 \)

- Simplified expression: \( 5y^3 - 3y^2 + 7y - 5 \)

Answer: \( 5y^3 - 3y^2 + 7y - 5 \)

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7. \( -(3a^2 + a + 2) + (-9a + 5) + 5a^2 \)

- Distribute the negative sign:
- \( -3a^2 - a - 2 - 9a + 5 + 5a^2 \)

- Combine like terms:
- \( a^2 \)-terms: \( -3a^2 + 5a^2 = 2a^2 \)
- \( a \)-terms: \( -a - 9a = -10a \)
- Constant terms: \( -2 + 5 = 3 \)

- Simplified expression: \( 2a^2 - 10a + 3 \)

Answer: \( 2a^2 - 10a + 3 \)

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8. \( (-4a^3 - a^2 + 8a) - (-9a^3 + 6a^2 - 3) \)

- Distribute the negative sign:
- \( -4a^3 - a^2 + 8a + 9a^3 - 6a^2 + 3 \)

- Combine like terms:
- \( a^3 \)-terms: \( -4a^3 + 9a^3 = 5a^3 \)
- \( a^2 \)-terms: \( -a^2 - 6a^2 = -7a^2 \)
- \( a \)-terms: \( 8a \) (no other \( a \)-terms)
- Constant terms: \( 3 \) (no other constant terms)

- Simplified expression: \( 5a^3 - 7a^2 + 8a + 3 \)

Answer: \( 5a^3 - 7a^2 + 8a + 3 \)

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9. \( -(3c^2 - 4c - 9) + (6c^2 + 3c + 7) \)

- Distribute the negative sign:
- \( -3c^2 + 4c + 9 + 6c^2 + 3c + 7 \)

- Combine like terms:
- \( c^2 \)-terms: \( -3c^2 + 6c^2 = 3c^2 \)
- \( c \)-terms: \( 4c + 3c = 7c \)
- Constant terms: \( 9 + 7 = 16 \)

- Simplified expression: \( 3c^2 + 7c + 16 \)

Answer: \( 3c^2 + 7c + 16 \)

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10. \( (-7x^3 + 5x^2 - 9) - (2x - 9) \)

- Distribute the negative sign:
- \( -7x^3 + 5x^2 - 9 - 2x + 9 \)

- Combine like terms:
- \( x^3 \)-terms: \( -7x^3 \) (no other \( x^3 \)-terms)
- \( x^2 \)-terms: \( 5x^2 \) (no other \( x^2 \)-terms)
- \( x \)-terms: \( -2x \) (no other \( x \)-terms)
- Constant terms: \( -9 + 9 = 0 \)

- Simplified expression: \( -7x^3 + 5x^2 - 2x \)

Answer: \( -7x^3 + 5x^2 - 2x \)

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11. \( (3y + 2) - 5y^2 + (3y^3 - y - 1) \)

- Rearrange and combine like terms:
- \( 3y^3 - 5y^2 + 3y - y + 2 - 1 \)

- Combine like terms:
- \( y^3 \)-terms: \( 3y^3 \) (no other \( y^3 \)-terms)
- \( y^2 \)-terms: \( -5y^2 \) (no other \( y^2 \)-terms)
- \( y \)-terms: \( 3y - y = 2y \)
- Constant terms: \( 2 - 1 = 1 \)

- Simplified expression: \( 3y^3 - 5y^2 + 2y + 1 \)

Answer: \( 3y^3 - 5y^2 + 2y + 1 \)

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12. \( (-12x^2 - 3x) + 5x - (x^2 - 2) \)

- Distribute the negative sign:
- \( -12x^2 - 3x + 5x - x^2 + 2 \)

- Combine like terms:
- \( x^2 \)-terms: \( -12x^2 - x^2 = -13x^2 \)
- \( x \)-terms: \( -3x + 5x = 2x \)
- Constant terms: \( 2 \) (no other constant terms)

- Simplified expression: \( -13x^2 + 2x + 2 \)

Answer: \( -13x^2 + 2x + 2 \)

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13. \( 7z^3 - (3x^2 + 3) + (-2x - x - 1) \)

- Note: This expression involves different variables (\( z \), \( x \)). Treat them separately.
- Simplify each part:
- \( 7z^3 \) (no other \( z^3 \)-terms)
- \( -3x^2 \) (no other \( x^2 \)-terms)
- \( -2x - x = -3x \)
- Constant terms: \( -3 - 1 = -4 \)

- Simplified expression: \( 7z^3 - 3x^2 - 3x - 4 \)

Answer: \( 7z^3 - 3x^2 - 3x - 4 \)

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14. \( 8m^2 + (-12m + 6) - (3m^3 + 1) + 5m^3 \)

- Distribute the negative sign:
- \( 8m^2 - 12m + 6 - 3m^3 - 1 + 5m^3 \)

- Combine like terms:
- \( m^3 \)-terms: \( -3m^3 + 5m^3 = 2m^3 \)
- \( m^2 \)-terms: \( 8m^2 \) (no other \( m^2 \)-terms)
- \( m \)-terms: \( -12m \) (no other \( m \)-terms)
- Constant terms: \( 6 - 1 = 5 \)

- Simplified expression: \( 2m^3 + 8m^2 - 12m + 5 \)

Answer: \( 2m^3 + 8m^2 - 12m + 5 \)

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15. \( (-6x + 2) - (3x - 1) - 2x^2 - (-3x^2 - 1) \)

- Distribute the negative signs:
- \( -6x + 2 - 3x + 1 - 2x^2 + 3x^2 + 1 \)

- Combine like terms:
- \( x^2 \)-terms: \( -2x^2 + 3x^2 = x^2 \)
- \( x \)-terms: \( -6x - 3x = -9x \)
- Constant terms: \( 2 + 1 + 1 = 4 \)

- Simplified expression: \( x^2 - 9x + 4 \)

Answer: \( x^2 - 9x + 4 \)

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16. \( (4a + 2) + (5a + 5) + a^2 + (2a^2 + 1) \)

- Combine like terms:
- \( a^2 \)-terms: \( a^2 + 2a^2 = 3a^2 \)
- \( a \)-terms: \( 4a + 5a = 9a \)
- Constant terms: \( 2 + 5 + 1 = 8 \)

- Simplified expression: \( 3a^2 + 9a + 8 \)

Answer: \( 3a^2 + 9a + 8 \)

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17. \( (12y^3 - 4y - 2) - 5y^3 - (3y^2 - 2) \)

- Distribute the negative signs:
- \( 12y^3 - 4y - 2 - 5y^3 - 3y^2 + 2 \)

- Combine like terms:
- \( y^3 \)-terms: \( 12y^3 - 5y^3 = 7y^3 \)
- \( y^2 \)-terms: \( -3y^2 \) (no other \( y^2 \)-terms)
- \( y \)-terms: \( -4y \) (no other \( y \)-terms)
- Constant terms: \( -2 + 2 = 0 \)

- Simplified expression: \( 7y^3 - 3y^2 - 4y \)

Answer: \( 7y^3 - 3y^2 - 4y \)

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18. \( -5c^2 + 3c - (-4c^3 + c^2 - 8) + 2c^3 + (c^2 - 5c + 7) \)

- Distribute the negative signs:
- \( -5c^2 + 3c + 4c^3 - c^2 + 8 + 2c^3 + c^2 - 5c + 7 \)

- Combine like terms:
- \( c^3 \)-terms: \( 4c^3 + 2c^3 = 6c^3 \)
- \( c^2 \)-terms: \( -5c^2 - c^2 + c^2 = -5c^2 \)
- \( c \)-terms: \( 3c - 5c = -2c \)
- Constant terms: \( 8 + 7 = 15 \)

- Simplified expression: \( 6c^3 - 5c^2 - 2c + 15 \)

Answer: \( 6c^3 - 5c^2 - 2c + 15 \)

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19. \( 8a^3 - (6a + 3) - 3a + 4 + (5a - 3) \)

- Distribute the negative signs:
- \( 8a^3 - 6a - 3 - 3a + 4 + 5a - 3 \)

- Combine like terms:
- \( a^3 \)-terms: \( 8a^3 \) (no other \( a^3 \)-terms)
- \( a \)-terms: \( -6a - 3a + 5a = -4a \)
- Constant terms: \( -3 + 4 - 3 = -2 \)

- Simplified expression: \( 8a^3 - 4a - 2 \)

Answer: \( 8a^3 - 4a - 2 \)

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20. \( (3x + 3) + (x^2 + 9x - 2) - 13x - (2x - 1) - x^2 \)

- Distribute the negative signs:
- \( 3x + 3 + x^2 + 9x - 2 - 13x - 2x + 1 - x^2 \)

- Combine like terms:
- \( x^2 \)-terms: \( x^2 - x^2 = 0 \)
- \( x \)-terms: \( 3x + 9x - 13x - 2x = -3x \)
- Constant terms: \( 3 - 2 + 1 = 2 \)

- Simplified expression: \( -3x + 2 \)

Answer: \( -3x + 2 \)

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Final Answers:

1. \( \boxed{8x^2 - x - 3} \)
2. \( \boxed{14x^2 + x + 1} \)
3. \( \boxed{10x^3 - x^2 + 4x + 3} \)
4. \( \boxed{5y^2 + 11} \)
5. \( \boxed{m^2 - 3m - 3} \)
6. \( \boxed{5y^3 - 3y^2 + 7y - 5} \)
7. \( \boxed{2a^2 - 10a + 3} \)
8. \( \boxed{5a^3 - 7a^2 + 8a + 3} \)
9. \( \boxed{3c^2 + 7c + 16} \)
10. \( \boxed{-7x^3 + 5x^2 - 2x} \)
11. \( \boxed{3y^3 - 5y^2 + 2y + 1} \)
12. \( \boxed{-13x^2 + 2x + 2} \)
13. \( \boxed{7z^3 - 3x^2 - 3x - 4} \)
14. \( \boxed{2m^3 + 8m^2 - 12m + 5} \)
15. \( \boxed{x^2 - 9x + 4} \)
16. \( \boxed{3a^2 + 9a + 8} \)
17. \( \boxed{7y^3 - 3y^2 - 4y} \)
18. \( \boxed{6c^3 - 5c^2 - 2c + 15} \)
19. \( \boxed{8a^3 - 4a - 2} \)
20. \( \boxed{-3x + 2} \)