Free Adding and Subtracting Polynomials Worksheets - Free Printable

Problem: Adding and Subtracting Polynomials

The task involves performing operations on polynomials by adding or subtracting like terms. Below, I will solve each problem step by step.

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#### 1. $(12y^2 + 17y - 4) + (9y^2 - 13y + 3)$

- Combine like terms:
- $y^2$ terms: $12y^2 + 9y^2 = 21y^2$
- $y$ terms: $17y - 13y = 4y$
- Constant terms: $-4 + 3 = -1$

- Result: $21y^2 + 4y - 1$

Answer: $\boxed{21y^2 + 4y - 1}$

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#### 2. $(2x^3 + 7x^2 + x) + (2x^2 - 4x - 12)$

- Combine like terms:
- $x^3$ terms: $2x^3$
- $x^2$ terms: $7x^2 + 2x^2 = 9x^2$
- $x$ terms: $x - 4x = -3x$
- Constant terms: $-12$

- Result: $2x^3 + 9x^2 - 3x - 12$

Answer: $\boxed{2x^3 + 9x^2 - 3x - 12}$

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#### 3. $(-3m^2 + m) + (4m^2 + 6m)$

- Combine like terms:
- $m^2$ terms: $-3m^2 + 4m^2 = m^2$
- $m$ terms: $m + 6m = 7m$

- Result: $m^2 + 7m$

Answer: $\boxed{m^2 + 7m}$

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#### 4. $(7z^3 + 4z - 1) + (2z^2 - 6z + 2)$

- Combine like terms:
- $z^3$ terms: $7z^3$
- $z^2$ terms: $2z^2$
- $z$ terms: $4z - 6z = -2z$
- Constant terms: $-1 + 2 = 1$

- Result: $7z^3 + 2z^2 - 2z + 1$

Answer: $\boxed{7z^3 + 2z^2 - 2z + 1}$

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

- Distribute the negative sign:
- $3a^2 + 2a - 2 - a^2 + 3a - 7$

- Combine like terms:
- $a^2$ terms: $3a^2 - a^2 = 2a^2$
- $a$ terms: $2a + 3a = 5a$
- Constant terms: $-2 - 7 = -9$

- Result: $2a^2 + 5a - 9$

Answer: $\boxed{2a^2 + 5a - 9}$

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#### 6. $(5x^2 - 2x - 1) - (3x^2 - 5x + 7)$

- Distribute the negative sign:
- $5x^2 - 2x - 1 - 3x^2 + 5x - 7$

- Combine like terms:
- $x^2$ terms: $5x^2 - 3x^2 = 2x^2$
- $x$ terms: $-2x + 5x = 3x$
- Constant terms: $-1 - 7 = -8$

- Result: $2x^2 + 3x - 8$

Answer: $\boxed{2x^2 + 3x - 8}$

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#### 7. $-(3z^2 + 4z) - (6z^2 - 2)$

- Distribute the negative signs:
- $-3z^2 - 4z - 6z^2 + 2$

- Combine like terms:
- $z^2$ terms: $-3z^2 - 6z^2 = -9z^2$
- $z$ terms: $-4z$
- Constant terms: $2$

- Result: $-9z^2 - 4z + 2$

Answer: $\boxed{-9z^2 - 4z + 2}$

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#### 8. $(6x^3 - 4x^2 + x - 9) - (3x^2 + 7x + 3)$

- Distribute the negative sign:
- $6x^3 - 4x^2 + x - 9 - 3x^2 - 7x - 3$

- Combine like terms:
- $x^3$ terms: $6x^3$
- $x^2$ terms: $-4x^2 - 3x^2 = -7x^2$
- $x$ terms: $x - 7x = -6x$
- Constant terms: $-9 - 3 = -12$

- Result: $6x^3 - 7x^2 - 6x - 12$

Answer: $\boxed{6x^3 - 7x^2 - 6x - 12}$

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#### 9. $(2x^2 + 1) + (x^2 - 2x + 1)$

- Combine like terms:
- $x^2$ terms: $2x^2 + x^2 = 3x^2$
- $x$ terms: $-2x$
- Constant terms: $1 + 1 = 2$

- Result: $3x^2 - 2x + 2$

Answer: $\boxed{3x^2 - 2x + 2}$

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#### 10. $(-s^2 - 3) - (2s^2 + 10s)$

- Distribute the negative sign:
- $-s^2 - 3 - 2s^2 - 10s$

- Combine like terms:
- $s^2$ terms: $-s^2 - 2s^2 = -3s^2$
- $s$ terms: $-10s$
- Constant terms: $-3$

- Result: $-3s^2 - 10s - 3$

Answer: $\boxed{-3s^2 - 10s - 3}$

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#### 11. $(5 - 9a^3) + (4a^2 + 6a - 3)$

- Combine like terms:
- $a^3$ terms: $-9a^3$
- $a^2$ terms: $4a^2$
- $a$ terms: $6a$
- Constant terms: $5 - 3 = 2$

- Result: $-9a^3 + 4a^2 + 6a + 2$

Answer: $\boxed{-9a^3 + 4a^2 + 6a + 2}$

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#### 12. $(3x^2 - x) + 5x^3 + (-4x^3 + x^2 - 8)$

- Combine like terms:
- $x^3$ terms: $5x^3 - 4x^3 = x^3$
- $x^2$ terms: $3x^2 + x^2 = 4x^2$
- $x$ terms: $-x$
- Constant terms: $-8$

- Result: $x^3 + 4x^2 - x - 8$

Answer: $\boxed{x^3 + 4x^2 - x - 8}$

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#### 13. $-10(u + v) + 8(u - 1) - 3(u + 6)$

- Distribute all terms:
- $-10u - 10v + 8u - 8 - 3u - 18$

- Combine like terms:
- $u$ terms: $-10u + 8u - 3u = -5u$
- $v$ terms: $-10v$
- Constant terms: $-8 - 18 = -26$

- Result: $-5u - 10v - 26$

Answer: $\boxed{-5u - 10v - 26}$

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#### 14. $7x - [2(x^2 - z) + 4x^2 - 7z] + 6z^2$

- Simplify inside the brackets:
- $2(x^2 - z) = 2x^2 - 2z$
- So, $2(x^2 - z) + 4x^2 - 7z = 2x^2 - 2z + 4x^2 - 7z = 6x^2 - 9z$

- Substitute back:
- $7x - (6x^2 - 9z) + 6z^2 = 7x - 6x^2 + 9z + 6z^2$

- Combine like terms:
- $x^2$ terms: $-6x^2$
- $x$ terms: $7x$
- $z^2$ terms: $6z^2$
- $z$ terms: $9z$

- Result: $-6x^2 + 7x + 6z^2 + 9z$

Answer: $\boxed{-6x^2 + 7x + 6z^2 + 9z}$

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#### 15. Subtract $t^4 - 3t^2 + 7$ from $5t^3 - 9$.

- Write the subtraction:
- $(5t^3 - 9) - (t^4 - 3t^2 + 7)$

- Distribute the negative sign:
- $5t^3 - 9 - t^4 + 3t^2 - 7$

- Combine like terms:
- $t^4$ terms: $-t^4$
- $t^3$ terms: $5t^3$
- $t^2$ terms: $3t^2$
- Constant terms: $-9 - 7 = -16$

- Result: $-t^4 + 5t^3 + 3t^2 - 16$

Answer: $\boxed{-t^4 + 5t^3 + 3t^2 - 16}$

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#### 16. Subtract $y^5 - y^4$ from $y^2 + 3y^4$.

- Write the subtraction:
- $(y^2 + 3y^4) - (y^5 - y^4)$

- Distribute the negative sign:
- $y^2 + 3y^4 - y^5 + y^4$

- Combine like terms:
- $y^5$ terms: $-y^5$
- $y^4$ terms: $3y^4 + y^4 = 4y^4$
- $y^2$ terms: $y^2$

- Result: $-y^5 + 4y^4 + y^2$

Answer: $\boxed{-y^5 + 4y^4 + y^2}$

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#### 17. Add $4(m^2 + 2)$ to $3m^2 + 7m$.

- Distribute the $4$:
- $4(m^2 + 2) = 4m^2 + 8$

- Add to $3m^2 + 7m$:
- $3m^2 + 7m + 4m^2 + 8$

- Combine like terms:
- $m^2$ terms: $3m^2 + 4m^2 = 7m^2$
- $m$ terms: $7m$
- Constant terms: $8$

- Result: $7m^2 + 7m + 8$

Answer: $\boxed{7m^2 + 7m + 8}$

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#### 18. $3(x^2 - 2x + 3) - 4(4x + 1) - (3x^2 - 2x)$

- Distribute all terms:
- $3(x^2 - 2x + 3) = 3x^2 - 6x + 9$
- $-4(4x + 1) = -16x - 4$
- $-(3x^2 - 2x) = -3x^2 + 2x$

- Combine all terms:
- $3x^2 - 6x + 9 - 16x - 4 - 3x^2 + 2x$

- Combine like terms:
- $x^2$ terms: $3x^2 - 3x^2 = 0$
- $x$ terms: $-6x - 16x + 2x = -20x$
- Constant terms: $9 - 4 = 5$

- Result: $-20x + 5$

Answer: $\boxed{-20x + 5}$

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#### 19. $(0.5x^2 + 4.25x - 0.9) - 0.5(x^2 + 7x - 3)$

- Distribute the $0.5$ in the second term:
- $0.5(x^2 + 7x - 3) = 0.5x^2 + 3.5x - 1.5$

- Subtract:
- $(0.5x^2 + 4.25x - 0.9) - (0.5x^2 + 3.5x - 1.5)$

- Distribute the negative sign:
- $0.5x^2 + 4.25x - 0.9 - 0.5x^2 - 3.5x + 1.5$

- Combine like terms:
- $x^2$ terms: $0.5x^2 - 0.5x^2 = 0$
- $x$ terms: $4.25x - 3.5x = 0.75x$
- Constant terms: $-0.9 + 1.5 = 0.6$

- Result: $0.75x + 0.6$

Answer: $\boxed{0.75x + 0.6}$

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#### 20. $0.23(4x^2 + 9x - 4) + 0.9(0.04x^2 + 3x - 7)$

- Distribute the constants:
- $0.23(4x^2 + 9x - 4) = 0.92x^2 + 2.07x - 0.92$
- $0.9(0.04x^2 + 3x - 7) = 0.036x^2 + 2.7x - 6.3$

- Add the results:
- $0.92x^2 + 2.07x - 0.92 + 0.036x^2 + 2.7x - 6.3$

- Combine like terms:
- $x^2$ terms: $0.92x^2 + 0.036x^2 = 0.956x^2$
- $x$ terms: $2.07x + 2.7x = 4.77x$
- Constant terms: $-0.92 - 6.3 = -7.22$

- Result: $0.956x^2 + 4.77x - 7.22$

Answer: $\boxed{0.956x^2 + 4.77x - 7.22}$

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Final Answers:

1. $\boxed{21y^2 + 4y - 1}$
2. $\boxed{2x^3 + 9x^2 - 3x - 12}$
3. $\boxed{m^2 + 7m}$
4. $\boxed{7z^3 + 2z^2 - 2z + 1}$
5. $\boxed{2a^2 + 5a - 9}$
6. $\boxed{2x^2 + 3x - 8}$
7. $\boxed{-9z^2 - 4z + 2}$
8. $\boxed{6x^3 - 7x^2 - 6x - 12}$
9. $\boxed{3x^2 - 2x + 2}$
10. $\boxed{-3s^2 - 10s - 3}$
11. $\boxed{-9a^3 + 4a^2 + 6a + 2}$
12. $\boxed{x^3 + 4x^2 - x - 8}$
13. $\boxed{-5u - 10v - 26}$
14. $\boxed{-6x^2 + 7x + 6z^2 + 9z}$
15. $\boxed{-t^4 + 5t^3 + 3t^2 - 16}$
16. $\boxed{-y^5 + 4y^4 + y^2}$
17. $\boxed{7m^2 + 7m + 8}$
18. $\boxed{-20x + 5}$
19. $\boxed{0.75x + 0.6}$
20. $\boxed{0.956x^2 + 4.77x - 7.22}$