Students can practice combining like terms and simplifying expressions with this adding and subtracting polynomials worksheet.

Let's solve each problem step by step, following the directions to add or subtract polynomials and write the answer in standard form (terms ordered from highest to lowest degree).

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1. $(5x^3 + 12) + (8x^3 - 7)$

Step 1: Remove parentheses:
$5x^3 + 12 + 8x^3 - 7$

Step 2: Combine like terms:
- $5x^3 + 8x^3 = 13x^3$
- $12 - 7 = 5$

Answer: $\boxed{13x^3 + 5}$

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2. $(6x^2 + 10x) - (4x^2 - 12x)$

Step 1: Distribute the negative sign:
$6x^2 + 10x - 4x^2 + 12x$

Step 2: Combine like terms:
- $6x^2 - 4x^2 = 2x^2$
- $10x + 12x = 22x$

Answer: $\boxed{2x^2 + 22x}$

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3. $(4y^4 + 10y^3 - 6) + (3y^4 - 7y^3 + 2y)$

Step 1: Remove parentheses:
$4y^4 + 10y^3 - 6 + 3y^4 - 7y^3 + 2y$

Step 2: Combine like terms:
- $4y^4 + 3y^4 = 7y^4$
- $10y^3 - 7y^3 = 3y^3$
- $2y$ stays
- $-6$ stays

Answer: $\boxed{7y^4 + 3y^3 + 2y - 6}$

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4. $(7x^2 - 5x + 4) + (6x^2 + 4x - 3)$

Step 1: Remove parentheses:
$7x^2 - 5x + 4 + 6x^2 + 4x - 3$

Step 2: Combine like terms:
- $7x^2 + 6x^2 = 13x^2$
- $-5x + 4x = -x$
- $4 - 3 = 1$

Answer: $\boxed{13x^2 - x + 1}$

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5. $(15x^3 - 3x^2 - 1) - (10x^3 - 4x^2 + 14)$

Step 1: Distribute the negative sign:
$15x^3 - 3x^2 - 1 - 10x^3 + 4x^2 - 14$

Step 2: Combine like terms:
- $15x^3 - 10x^3 = 5x^3$
- $-3x^2 + 4x^2 = x^2$
- $-1 - 14 = -15$

Answer: $\boxed{5x^3 + x^2 - 15}$

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6. $(3y^3 + 6y^2 + y) - (8y^3 - 4y^2 - y + 7)$

Step 1: Distribute the negative sign:
$3y^3 + 6y^2 + y - 8y^3 + 4y^2 + y - 7$

Step 2: Combine like terms:
- $3y^3 - 8y^3 = -5y^3$
- $6y^2 + 4y^2 = 10y^2$
- $y + y = 2y$
- $-7$

Answer: $\boxed{-5y^3 + 10y^2 + 2y - 7}$

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7. $(6y^3 + 10y) + (15y^3 + 2y^2 - 16y + 4)$

Step 1: Remove parentheses:
$6y^3 + 10y + 15y^3 + 2y^2 - 16y + 4$

Step 2: Combine like terms:
- $6y^3 + 15y^3 = 21y^3$
- $2y^2$ stays
- $10y - 16y = -6y$
- $+4$

Answer: $\boxed{21y^3 + 2y^2 - 6y + 4}$

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8. $(8x^3 + 2x^2) - (14x^3 - 10x^2 + 9)$

Step 1: Distribute the negative sign:
$8x^3 + 2x^2 - 14x^3 + 10x^2 - 9$

Step 2: Combine like terms:
- $8x^3 - 14x^3 = -6x^3$
- $2x^2 + 10x^2 = 12x^2$
- $-9$

Answer: $\boxed{-6x^3 + 12x^2 - 9}$

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9. Find the expression that represents the perimeter of the triangle.

The sides are:
- $5x + 13$
- $8x - 2$
- $3x + 10$

Perimeter = sum of all sides

Add them together:

$$
(5x + 13) + (8x - 2) + (3x + 10)
$$

Step 1: Remove parentheses:
$5x + 13 + 8x - 2 + 3x + 10$

Step 2: Combine like terms:
- $5x + 8x + 3x = 16x$
- $13 - 2 + 10 = 21$

Answer: $\boxed{16x + 21}$

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✔ Final Answers:

1. $13x^3 + 5$
2. $2x^2 + 22x$
3. $7y^4 + 3y^3 + 2y - 6$
4. $13x^2 - x + 1$
5. $5x^3 + x^2 - 15$
6. $-5y^3 + 10y^2 + 2y - 7$
7. $21y^3 + 2y^2 - 6y + 4$
8. $-6x^3 + 12x^2 - 9$
9. $16x + 21$

Let me know if you'd like this formatted as a printable worksheet!