Practice simplifying polynomial expressions with this worksheet containing 27 addition and subtraction problems.

Let's solve each of these polynomial addition and subtraction problems step by step. The goal is to simplify each expression by combining like terms. Like terms are terms that have the same variables raised to the same powers.

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🔷 Key Rules:

- When adding or subtracting polynomials, only combine like terms.
- Remember:
$$
(A) - (B) = A - B
$$
Distribute the negative sign to every term inside the parentheses when subtracting.

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Let’s go through each problem:

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#### 1) $(-4x^3y^3 - 6x^2y^2) - (2x^3y^3 - 7x^2y^2)$

Distribute the negative:
$$
= -4x^3y^3 - 6x^2y^2 - 2x^3y^3 + 7x^2y^2
$$
Combine like terms:
- $-4x^3y^3 - 2x^3y^3 = -6x^3y^3$
- $-6x^2y^2 + 7x^2y^2 = x^2y^2$

✔ Answer: $-6x^3y^3 + x^2y^2$

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#### 2) $(7y + 3x^2y^3) + (6x^2 + 4x^2y^3)$

Group like terms:
- $7y$ → no other $y$ term
- $3x^2y^3 + 4x^2y^3 = 7x^2y^3$
- $6x^2$ → no other $x^2$ term

✔ Answer: $7y + 7x^2y^3 + 6x^2$

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#### 3) $(-5x^2y - 4x^3) - (4x^3 + 6x^2y)$

Distribute the negative:
$$
= -5x^2y - 4x^3 - 4x^3 - 6x^2y
$$
Combine:
- $-5x^2y - 6x^2y = -11x^2y$
- $-4x^3 - 4x^3 = -8x^3$

✔ Answer: $-8x^3 - 11x^2y$

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#### 4) $(6x^2y^2 - 6x^3y) + (4x^3y - 4x^2y^2)$

Add:
- $6x^2y^2 - 4x^2y^2 = 2x^2y^2$
- $-6x^3y + 4x^3y = -2x^3y$

✔ Answer: $2x^2y^2 - 2x^3y$

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#### 5) $(5b^2 - 1) - (-8b - 7)$

Distribute negative:
$$
= 5b^2 - 1 + 8b + 7
$$
Combine:
- $5b^2$
- $8b$
- $-1 + 7 = 6$

✔ Answer: $5b^2 + 8b + 6$

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#### 6) $(-mn^3 + 2m^2) - (5mn^3 - 2m^2n^2)$

Distribute:
$$
= -mn^3 + 2m^2 - 5mn^3 + 2m^2n^2
$$
Combine:
- $-mn^3 - 5mn^3 = -6mn^3$
- $2m^2$
- $+2m^2n^2$

✔ Answer: $-6mn^3 + 2m^2 + 2m^2n^2$

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#### 7) $(-7xy^2 - 8) + (7 + 8x^3y^3)$

Add:
- $-7xy^2$
- $-8 + 7 = -1$
- $+8x^3y^3$

✔ Answer: $-7xy^2 + 8x^3y^3 - 1$

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

Add:
- $4x^2y^3 - 4x^2y^3 = 0$
- $-5x^2 - 4x^2 = -9x^2$

✔ Answer: $-9x^2$

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#### 9) $(4m^2 - 6m^3n^3) - (-8m^3n^3 + 5m^2)$

Distribute:
$$
= 4m^2 - 6m^3n^3 + 8m^3n^3 - 5m^2
$$
Combine:
- $4m^2 - 5m^2 = -m^2$
- $-6m^3n^3 + 8m^3n^3 = 2m^3n^3$

✔ Answer: $2m^3n^3 - m^2$

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#### 10) $(-uv - 7u^2v^2) + (-2uv + 4u^2v^2)$

Add:
- $-uv - 2uv = -3uv$
- $-7u^2v^2 + 4u^2v^2 = -3u^2v^2$

✔ Answer: $-3uv - 3u^2v^2$

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#### 11) $(6x - 6y) - (2x + 8y)$

Distribute:
$$
= 6x - 6y - 2x - 8y
$$
Combine:
- $6x - 2x = 4x$
- $-6y - 8y = -14y$

✔ Answer: $4x - 14y$

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#### 12) $(3v^3 - u^3v) + (7v^3 + u^3v)$

Add:
- $3v^3 + 7v^3 = 10v^3$
- $-u^3v + u^3v = 0$

✔ Answer: $10v^3$

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#### 13) $(-5x^2y + 3x^3y) - (-5x^3y + x^2y)$

Distribute:
$$
= -5x^2y + 3x^3y + 5x^3y - x^2y
$$
Combine:
- $3x^3y + 5x^3y = 8x^3y$
- $-5x^2y - x^2y = -6x^2y$

✔ Answer: $8x^3y - 6x^2y$

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#### 14) $(-4a^2b - a) - (5a - 3a^2b)$

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

✔ Answer: $-a^2b - 6a$

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#### 15) $(-7xy - 5y^2) + (5y^2 - 2xy)$

Add:
- $-7xy - 2xy = -9xy$
- $-5y^2 + 5y^2 = 0$

✔ Answer: $-9xy$

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#### 16) $(7m^3n - mn^2) - (-7m^3n + 8mn^2)$

Distribute:
$$
= 7m^3n - mn^2 + 7m^3n - 8mn^2
$$
Combine:
- $7m^3n + 7m^3n = 14m^3n$
- $-mn^2 - 8mn^2 = -9mn^2$

✔ Answer: $14m^3n - 9mn^2$

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#### 17) $(4x - 5y^2) + (-x + y^2)$

Add:
- $4x - x = 3x$
- $-5y^2 + y^2 = -4y^2$

✔ Answer: $3x - 4y^2$

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#### 18) $(8m^3n^2 + 4n^2) - (6n^2 - 6m^3n^2)$

Distribute:
$$
= 8m^3n^2 + 4n^2 - 6n^2 + 6m^3n^2
$$
Combine:
- $8m^3n^2 + 6m^3n^2 = 14m^3n^2$
- $4n^2 - 6n^2 = -2n^2$

✔ Answer: $14m^3n^2 - 2n^2$

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#### 19) $(-3u^3v^2 - 5u^3v) + (-8u^3v^2 + 4u^3v)$

Add:
- $-3u^3v^2 - 8u^3v^2 = -11u^3v^2$
- $-5u^3v + 4u^3v = -u^3v$

✔ Answer: $-11u^3v^2 - u^3v$

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#### 20) $(7v^2 - 5uv^2) - (2uv^2 + 7v^2)$

Distribute:
$$
= 7v^2 - 5uv^2 - 2uv^2 - 7v^2
$$
Combine:
- $7v^2 - 7v^2 = 0$
- $-5uv^2 - 2uv^2 = -7uv^2$

✔ Answer: $-7uv^2$

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#### 21) $(-6xy^3 + 2x^2y^2) + (-3xy^3 + 7x^2y^2)$

Add:
- $-6xy^3 - 3xy^3 = -9xy^3$
- $2x^2y^2 + 7x^2y^2 = 9x^2y^2$

✔ Answer: $-9xy^3 + 9x^2y^2$

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#### 22) $(-4xy^3 + 6x^3) - (8x^3 - 3xy^3)$

Distribute:
$$
= -4xy^3 + 6x^3 - 8x^3 + 3xy^3
$$
Combine:
- $-4xy^3 + 3xy^3 = -xy^3$
- $6x^3 - 8x^3 = -2x^3$

✔ Answer: $-xy^3 - 2x^3$

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#### 23) $(-7b^3 - 4a^2b^3) - (-5b^3 - 7a^2b^3)$

Distribute:
$$
= -7b^3 - 4a^2b^3 + 5b^3 + 7a^2b^3
$$
Combine:
- $-7b^3 + 5b^3 = -2b^3$
- $-4a^2b^3 + 7a^2b^3 = 3a^2b^3$

✔ Answer: $-2b^3 + 3a^2b^3$

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#### 24) $(7x^3y^3 + y^2) - (3y^2 - x^3y^3)$

Distribute:
$$
= 7x^3y^3 + y^2 - 3y^2 + x^3y^3
$$
Combine:
- $7x^3y^3 + x^3y^3 = 8x^3y^3$
- $y^2 - 3y^2 = -2y^2$

✔ Answer: $8x^3y^3 - 2y^2$

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#### 25) $(3m^3 - 4m^3n^3) - (5m^3n^3 - 4m^3)$

Distribute:
$$
= 3m^3 - 4m^3n^3 - 5m^3n^3 + 4m^3
$$
Combine:
- $3m^3 + 4m^3 = 7m^3$
- $-4m^3n^3 - 5m^3n^3 = -9m^3n^3$

✔ Answer: $7m^3 - 9m^3n^3$

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#### 26) $(7x + 7x^3y^3) + (-7x^3y^3 - 7x) - (6x^3y^2 - 2x)$

Break it into parts:

First: $7x + 7x^3y^3 - 7x^3y^3 - 7x = (7x - 7x) + (7x^3y^3 - 7x^3y^3) = 0$

Now subtract $(6x^3y^2 - 2x)$:
$$
0 - (6x^3y^2 - 2x) = -6x^3y^2 + 2x
$$

✔ Answer: $-6x^3y^2 + 2x$

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#### 27) $(-2u + 5v) - (-2v + 2u) - (7u - 7uv)$

Step-by-step:

First: $(-2u + 5v) - (-2v + 2u) = -2u + 5v + 2v - 2u = (-2u - 2u) + (5v + 2v) = -4u + 7v$

Now subtract $(7u - 7uv)$:
$$
(-4u + 7v) - (7u - 7uv) = -4u + 7v - 7u + 7uv = (-4u - 7u) + 7v + 7uv = -11u + 7v + 7uv
$$

✔ Answer: $-11u + 7v + 7uv$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $-6x^3y^3 + x^2y^2$ |
| 2 | $7y + 7x^2y^3 + 6x^2$ |
| 3 | $-8x^3 - 11x^2y$ |
| 4 | $2x^2y^2 - 2x^3y$ |
| 5 | $5b^2 + 8b + 6$ |
| 6 | $-6mn^3 + 2m^2 + 2m^2n^2$ |
| 7 | $-7xy^2 + 8x^3y^3 - 1$ |
| 8 | $-9x^2$ |
| 9 | $2m^3n^3 - m^2$ |
| 10 | $-3uv - 3u^2v^2$ |
| 11 | $4x - 14y$ |
| 12 | $10v^3$ |
| 13 | $8x^3y - 6x^2y$ |
| 14 | $-a^2b - 6a$ |
| 15 | $-9xy$ |
| 16 | $14m^3n - 9mn^2$ |
| 17 | $3x - 4y^2$ |
| 18 | $14m^3n^2 - 2n^2$ |
| 19 | $-11u^3v^2 - u^3v$ |
| 20 | $-7uv^2$ |
| 21 | $-9xy^3 + 9x^2y^2$ |
| 22 | $-xy^3 - 2x^3$ |
| 23 | $-2b^3 + 3a^2b^3$ |
| 24 | $8x^3y^3 - 2y^2$ |
| 25 | $7m^3 - 9m^3n^3$ |
| 26 | $-6x^3y^2 + 2x$ |
| 27 | $-11u + 7v + 7uv$ |

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