Simplifying Expressions worksheet with algebra problems for students to solve.

Let's go through each expression and simplify it by combining like terms. Like terms are terms that have the same variable raised to the same power.

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1. $ x + 3x - 4x^2 - x $

- Combine $ x + 3x - x = 3x $
- $ -4x^2 $ stays as is
- Result: $ 3x - 4x^2 $

✔ Answer: $ -4x^2 + 3x $

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2. $ x + 2x^2 - 4x^2 - 5x^2 $

- Combine $ 2x^2 - 4x^2 - 5x^2 = (2 - 4 - 5)x^2 = -7x^2 $
- $ x $ remains
- Result: $ x - 7x^2 $

✔ Answer: $ -7x^2 + x $

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3. $ 4 - d^2 + 2d^2 - d $

- $ -d^2 + 2d^2 = d^2 $
- $ -d $ stays
- $ 4 $ stays
- Result: $ 4 + d^2 - d $

✔ Answer: $ d^2 - d + 4 $

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4. $ c^2 + y - 5y + c^2 $

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

✔ Answer: $ 2c^2 - 4y $

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5. $ 1 - xy - 2 - xy $

- $ 1 - 2 = -1 $
- $ -xy - xy = -2xy $
- Result: $ -1 - 2xy $

✔ Answer: $ -2xy - 1 $

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6. $ 2x^2 + x + 4x - 3x $

- $ x + 4x - 3x = 2x $
- $ 2x^2 $ stays
- Result: $ 2x^2 + 2x $

✔ Answer: $ 2x^2 + 2x $

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7. $ u - 3bu + 1 - bu $

- $ u $ has no like term
- $ -3bu - bu = -4bu $
- $ 1 $ stays
- Result: $ u - 4bu + 1 $

✔ Answer: $ -4bu + u + 1 $

> Note: $ u $ and $ bu $ are not like terms unless $ b $ is a constant.

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8. $ a^2 + 2 + a - 2a^2 $

- $ a^2 - 2a^2 = -a^2 $
- $ a $ stays
- $ 2 $ stays
- Result: $ -a^2 + a + 2 $

✔ Answer: $ -a^2 + a + 2 $

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9. $ -2y^2 - 2y^2 + y + 2y $

- $ -2y^2 - 2y^2 = -4y^2 $
- $ y + 2y = 3y $
- Result: $ -4y^2 + 3y $

✔ Answer: $ -4y^2 + 3y $

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10. $ r + r^2 - r $

- $ r - r = 0 $
- $ r^2 $ stays
- Result: $ r^2 $

✔ Answer: $ r^2 $

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11. $ 6x^2 - 5 + 6 $

- $ -5 + 6 = 1 $
- $ 6x^2 $ stays
- Result: $ 6x^2 + 1 $

✔ Answer: $ 6x^2 + 1 $

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12. $ 8 + 10x^2 - x^2 $

- $ 10x^2 - x^2 = 9x^2 $
- $ 8 $ stays
- Result: $ 9x^2 + 8 $

✔ Answer: $ 9x^2 + 8 $

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13. $ r^2 - r + 3r - r^2 $

- $ r^2 - r^2 = 0 $
- $ -r + 3r = 2r $
- Result: $ 2r $

✔ Answer: $ 2r $

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14. $ t^2 - 5t + 5t - t^2 $

- $ t^2 - t^2 = 0 $
- $ -5t + 5t = 0 $
- Result: $ 0 $

✔ Answer: $ 0 $

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15. $ 5 + w + 2w^2 - w^2 $

- $ 2w^2 - w^2 = w^2 $
- $ w $ stays
- $ 5 $ stays
- Result: $ w^2 + w + 5 $

✔ Answer: $ w^2 + w + 5 $

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16. $ 3r - 6 + r + 3r^2 $

- $ 3r + r = 4r $
- $ -6 $ stays
- $ 3r^2 $ stays
- Result: $ 3r^2 + 4r - 6 $

✔ Answer: $ 3r^2 + 4r - 6 $

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17. $ 3a + 3a^2 - 4a^2 + 5a $

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

✔ Answer: $ -a^2 + 8a $

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18. $ 5q^3 + 5q^2 - 2q^3 - 2q^2 - 3q^2 $

Wait — this seems to be two lines:

From image:
- $ 5q^3 + 5q^2 - 2q^3 - 2q^2 $
- Then below: $ 3q^3 - 3q^2 $

But likely the last one is part of the same problem? Let's assume the full expression is:

$ 5q^3 + 5q^2 - 2q^3 - 2q^2 - 3q^3 - 3q^2 $

(Combining both lines)

So:
- $ 5q^3 - 2q^3 - 3q^3 = (5 - 2 - 3)q^3 = 0q^3 $
- $ 5q^2 - 2q^2 - 3q^2 = (5 - 2 - 3)q^2 = 0q^2 $

✔ Answer: $ 0 $

> Alternatively, if only the first line was intended:
> $ 5q^3 + 5q^2 - 2q^3 - 2q^2 = (5 - 2)q^3 + (5 - 2)q^2 = 3q^3 + 3q^2 $

But since there's a second line written in red: $ 3q^3 - 3q^2 $, it might be a separate expression?

Let’s re-check.

Looking at layout:

```
5q^3 + 5q^2 - 2q^3 - 2q^2
3q^3 - 3q^2
```

Possibly two expressions?

But formatting suggests they're on the same row. Likely typo or misalignment.

Assuming the full expression is:

$ 5q^3 + 5q^2 - 2q^3 - 2q^2 - 3q^3 - 3q^2 $

Then:
- $ q^3 $: $ 5 - 2 - 3 = 0 $
- $ q^2 $: $ 5 - 2 - 3 = 0 $

✔ Answer: $ 0 $

Alternatively, if the second line is a different expression: $ 3q^3 - 3q^2 $, then simplify as is.

But based on context, likely the entire block is one expression.

We'll go with $ 0 $ for now.

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19. $ -r - 6 + 2r - 9 $

- $ -r + 2r = r $
- $ -6 - 9 = -15 $
- Result: $ r - 15 $

✔ Answer: $ r - 15 $

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20. $ u^3 + 4u + u^2 - 2u $

- $ 4u - 2u = 2u $
- $ u^3 $, $ u^2 $ stay
- Result: $ u^3 + u^2 + 2u $

✔ Answer: $ u^3 + u^2 + 2u $

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21. $ 1 + av + 3 - 3a $

- $ 1 + 3 = 4 $
- $ av $ stays
- $ -3a $ stays
- $ av $ and $ a $ are not like terms

Result: $ av - 3a + 4 $

✔ Answer: $ av - 3a + 4 $

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## ✔ Final Answers (Simplified Expressions):

| Expression | Simplified |
|----------|------------|
| 1. $ x + 3x - 4x^2 - x $ | $ -4x^2 + 3x $ |
| 2. $ x + 2x^2 - 4x^2 - 5x^2 $ | $ -7x^2 + x $ |
| 3. $ 4 - d^2 + 2d^2 - d $ | $ d^2 - d + 4 $ |
| 4. $ c^2 + y - 5y + c^2 $ | $ 2c^2 - 4y $ |
| 5. $ 1 - xy - 2 - xy $ | $ -2xy - 1 $ |
| 6. $ 2x^2 + x + 4x - 3x $ | $ 2x^2 + 2x $ |
| 7. $ u - 3bu + 1 - bu $ | $ -4bu + u + 1 $ |
| 8. $ a^2 + 2 + a - 2a^2 $ | $ -a^2 + a + 2 $ |
| 9. $ -2y^2 - 2y^2 + y + 2y $ | $ -4y^2 + 3y $ |
| 10. $ r + r^2 - r $ | $ r^2 $ |
| 11. $ 6x^2 - 5 + 6 $ | $ 6x^2 + 1 $ |
| 12. $ 8 + 10x^2 - x^2 $ | $ 9x^2 + 8 $ |
| 13. $ r^2 - r + 3r - r^2 $ | $ 2r $ |
| 14. $ t^2 - 5t + 5t - t^2 $ | $ 0 $ |
| 15. $ 5 + w + 2w^2 - w^2 $ | $ w^2 + w + 5 $ |
| 16. $ 3r - 6 + r + 3r^2 $ | $ 3r^2 + 4r - 6 $ |
| 17. $ 3a + 3a^2 - 4a^2 + 5a $ | $ -a^2 + 8a $ |
| 18. $ 5q^3 + 5q^2 - 2q^3 - 2q^2 - 3q^3 - 3q^2 $ | $ 0 $ |
| 19. $ -r - 6 + 2r - 9 $ | $ r - 15 $ |
| 20. $ u^3 + 4u + u^2 - 2u $ | $ u^3 + u^2 + 2u $ |
| 21. $ 1 + av + 3 - 3a $ | $ av - 3a + 4 $ |

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🔍 Explanation Summary:

To simplify algebraic expressions:
1. Identify like terms (same variables and exponents).
2. Combine coefficients using addition/subtraction.
3. Keep unlike terms separate.
4. Write in standard form (descending powers).

For example:
- $ 3x + 2x = 5x $
- $ 4x^2 - x^2 = 3x^2 $
- $ 2x $ and $ 3y $ cannot be combined.

This method applies to all problems above.

Let me know if you'd like these written in a printable format!