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

Let's solve each of these polynomial addition and subtraction problems step by step. The key idea is to combine like terms — terms with the same variable and exponent.

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

Add like terms:
- $5x^2 + 3x^2 = 8x^2$
- $-2x + x = -x$
- $-2 - 1 = -3$

✔ Answer: $8x^2 - x - 3$

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

- $2x^2 + 12x^2 = 14x^2$
- $5x - 4x = x$
- $-3 + 4 = 1$

✔ Answer: $14x^2 + x + 1$

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

- $10x^3$ (only one)
- $-6x^2 + 5x^2 = -x^2$
- $2x + 2x = 4x$
- $+3$

✔ Answer: $10x^3 - x^2 + 4x + 3$

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

Distribute the minus sign:
- $7y^2 - 2y^2 = 5y^2$
- $4y - 4y = 0$
- $-1 - (-12) = -1 + 12 = 11$

✔ Answer: $5y^2 + 11$

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

Distribute minus:
- $3m^2 - 2m^2 = m^2$
- $-2m - m = -3m$
- $-2 - 1 = -3$

✔ Answer: $m^2 - 3m - 3$

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

Distribute minus:
- $5y^3$ (only one)
- $-3y^2$ (no other $y^2$ term)
- $2y - (-5y) = 2y + 5y = 7y$
- $-12 - (-7) = -12 + 7 = -5$

✔ Answer: $5y^3 - 3y^2 + 7y - 5$

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

First distribute the negative:
- $-3a^2 - a - 2$
Now add all:
- $-3a^2 + 5a^2 = 2a^2$
- $-a - 9a = -10a$
- $-2 + 5 = 3$

✔ Answer: $2a^2 - 10a + 3$

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

Distribute minus:
- $-4a^3 + 9a^3 = 5a^3$
- $-a^2 - 6a^2 = -7a^2$
- $8a - (-3) = 8a + 3$

Wait: $8a$ stays, and we subtract $-3$, so $+3$

So: $5a^3 - 7a^2 + 8a + 3$

✔ Answer: $5a^3 - 7a^2 + 8a + 3$

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

Distribute negative:
- $-3c^2 + 4c + 9$
Now add:
- $-3c^2 + 6c^2 = 3c^2$
- $4c + 3c = 7c$
- $9 + 7 = 16$

✔ Answer: $3c^2 + 7c + 16$

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

Distribute minus:
- $-7x^3 + 5x^2 - 9 - 2x + 9$
- $-7x^3 + 5x^2 - 2x + (-9 + 9) = -7x^3 + 5x^2 - 2x$

✔ Answer: $-7x^3 + 5x^2 - 2x$

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

Rewrite: $3y + 2 - 5y^2 + 3y^3 - y - 1$

Group like terms:
- $3y^3$
- $-5y^2$
- $3y - y = 2y$
- $2 - 1 = 1$

✔ Answer: $3y^3 - 5y^2 + 2y + 1$

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

Simplify step by step:
- First: $-12x^2 - 3x + 5x = -12x^2 + 2x$
- Now subtract $(x^2 - 2)$ → $-x^2 + 2$
- So: $-12x^2 + 2x - x^2 + 2 = -13x^2 + 2x + 2$

✔ Answer: $-13x^2 + 2x + 2$

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

Note: There are different variables — $z$ and $x$. No like terms between them.

So:
- $7z^3$
- $-3x^2$
- $-2x - x = -3x$
- $-3 - 1 = -4$

✔ Answer: $7z^3 - 3x^2 - 3x - 4$

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

Break it down:
- $8m^2$
- $-12m + 6$
- $-3m^3 - 1$
- $+5m^3$

Now combine:
- $-3m^3 + 5m^3 = 2m^3$
- $8m^2$
- $-12m$
- $6 - 1 = 5$

✔ Answer: $2m^3 + 8m^2 - 12m + 5$

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

Step-by-step:
- $-6x + 2 - 3x + 1 - 2x^2 + 3x^2 + 1$

Now group:
- $-2x^2 + 3x^2 = x^2$
- $-6x - 3x = -9x$
- $2 + 1 + 1 = 4$

✔ Answer: $x^2 - 9x + 4$

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

Group:
- $a^2 + 2a^2 = 3a^2$
- $4a + 5a = 9a$
- $2 + 5 + 1 = 8$

✔ Answer: $3a^2 + 9a + 8$

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

Break down:
- $12y^3 - 5y^3 = 7y^3$
- $-4y$
- $-2$
- $-3y^2 + 2$ (because minus the whole thing)

So:
- $7y^3$
- $-3y^2$
- $-4y$
- $-2 + 2 = 0$

✔ 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 minus:
- $-5c^2 + 3c + 4c^3 - c^2 + 8 + 2c^3 + c^2 - 5c + 7$

Now combine:
- $4c^3 + 2c^3 = 6c^3$
- $-5c^2 - c^2 + c^2 = -5c^2$
- $3c - 5c = -2c$
- $8 + 7 = 15$

✔ Answer: $6c^3 - 5c^2 - 2c + 15$

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

Distribute minus:
- $8a^3 - 6a - 3 - 3a + 4 + 5a - 3$

Combine:
- $8a^3$
- $-6a - 3a + 5a = -4a$
- $-3 + 4 - 3 = -2$

✔ Answer: $8a^3 - 4a - 2$

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

Break it down:
- $3x + 3 + x^2 + 9x - 2 - 13x - 2x + 1 - x^2$

Now combine:
- $x^2 - x^2 = 0$
- $3x + 9x - 13x - 2x = (12x - 15x) = -3x$
- $3 - 2 + 1 = 2$

✔ Answer: $-3x + 2$

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✔ Final Answers:

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

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