Adding and Subtracting Polynomials worksheet with 15 algebraic expressions to simplify.

Sure! Let’s simplify each polynomial expression step by step. The key idea is to combine like terms — terms that have the same variable(s) raised to the same power.

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1) `(12y² + 17y − 4) + (9y² − 13y + 3)`

Combine like terms:

- y² terms: `12y² + 9y² = 21y²`
- y terms: `17y − 13y = 4y`
- constants: `−4 + 3 = −1`

✔ Answer: `21y² + 4y − 1`

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2) `(−3m² + m) + (4m² + 6m)`

- m² terms: `−3m² + 4m² = m²`
- m terms: `m + 6m = 7m`

✔ Answer: `m² + 7m`

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3) `(2x² + 1) + (x² − 2x + 1)`

- x² terms: `2x² + x² = 3x²`
- x terms: `−2x` (only one)
- constants: `1 + 1 = 2`

✔ Answer: `3x² − 2x + 2`

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4) `(4r³ + 3r⁴) − (r⁴ − 5r³)`

Distribute the minus sign:

= `4r³ + 3r⁴ − r⁴ + 5r³`

Now combine:

- r⁴ terms: `3r⁴ − r⁴ = 2r⁴`
- r³ terms: `4r³ + 5r³ = 9r³`

✔ Answer: `2r⁴ + 9r³`

*(Note: Usually we write in descending order → `2r⁴ + 9r³`)*

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5) `(4n − 3n³) − (3n³ + 4n)`

Distribute minus:

= `4n − 3n³ − 3n³ − 4n`

Combine:

- n³ terms: `−3n³ − 3n³ = −6n³`
- n terms: `4n − 4n = 0`

✔ Answer: `−6n³`

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6) `(−x⁴ + 13x⁵ + 6x³) + (6x³ + 5x⁵ + 7x⁴)`

Group like terms:

- x⁵: `13x⁵ + 5x⁵ = 18x⁵`
- x⁴: `−x⁴ + 7x⁴ = 6x⁴`
- x³: `6x³ + 6x³ = 12x³`

✔ Answer: `18x⁵ + 6x⁴ + 12x³`

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7) `(−x² + x − 4) − (3x² − 8x − 2)`

Distribute minus:

= `−x² + x − 4 − 3x² + 8x + 2`

Combine:

- x²: `−x² − 3x² = −4x²`
- x: `x + 8x = 9x`
- constants: `−4 + 2 = −2`

✔ Answer: `−4x² + 9x − 2`

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8) `(−2x³ + x) − (7x − 3 − 7x³)`

Distribute minus:

= `−2x³ + x − 7x + 3 + 7x³`

Combine:

- x³: `−2x³ + 7x³ = 5x³`
- x: `x − 7x = −6x`
- constant: `+3`

✔ Answer: `5x³ − 6x + 3`

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9) `(5x³ + 5x² + 5) − (6x³ − 6x² + 8x − 5)`

Distribute minus:

= `5x³ + 5x² + 5 − 6x³ + 6x² − 8x + 5`

Combine:

- x³: `5x³ − 6x³ = −x³`
- x²: `5x² + 6x² = 11x²`
- x: `−8x`
- constants: `5 + 5 = 10`

✔ Answer: `−x³ + 11x² − 8x + 10`

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10) `(5x³ + 3x² + 5) − (7x³ − 9x² + 8x − 5)`

Distribute minus:

= `5x³ + 3x² + 5 − 7x³ + 9x² − 8x + 5`

Combine:

- x³: `5x³ − 7x³ = −2x³`
- x²: `3x² + 9x² = 12x²`
- x: `−8x`
- constants: `5 + 5 = 10`

✔ Answer: `−2x³ + 12x² − 8x + 10`

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11) `7x⁴ + x − 2 − 7 − 5x² − 4x⁴`

Group like terms:

- x⁴: `7x⁴ − 4x⁴ = 3x⁴`
- x²: `−5x²`
- x: `x`
- constants: `−2 − 7 = −9`

✔ Answer: `3x⁴ − 5x² + x − 9`

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12) `(6x⁴ − 3 − 8x) + (6 + 4x + 8x⁴)`

Combine:

- x⁴: `6x⁴ + 8x⁴ = 14x⁴`
- x: `−8x + 4x = −4x`
- constants: `−3 + 6 = 3`

✔ Answer: `14x⁴ − 4x + 3`

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13) `(3x⁴ − 3x) − (3x − 3x⁴)`

Distribute minus:

= `3x⁴ − 3x − 3x + 3x⁴`

Combine:

- x⁴: `3x⁴ + 3x⁴ = 6x⁴`
- x: `−3x − 3x = −6x`

✔ Answer: `6x⁴ − 6x`

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14) `(3 − 6n⁵ − 8n⁴) − (−6n⁴ − 3n − 8n⁵)`

Distribute minus:

= `3 − 6n⁵ − 8n⁴ + 6n⁴ + 3n + 8n⁵`

Combine:

- n⁵: `−6n⁵ + 8n⁵ = 2n⁵`
- n⁴: `−8n⁴ + 6n⁴ = −2n⁴`
- n: `+3n`
- constant: `+3`

✔ Answer: `2n⁵ − 2n⁴ + 3n + 3`

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15) `(8n − 3n⁴ + 10n²) − (3n² + 11n⁴ − 7)`

Distribute minus:

= `8n − 3n⁴ + 10n² − 3n² − 11n⁴ + 7`

Combine:

- n⁴: `−3n⁴ − 11n⁴ = −14n⁴`
- n²: `10n² − 3n² = 7n²`
- n: `8n`
- constant: `+7`

✔ Answer: `−14n⁴ + 7n² + 8n + 7`

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## ✔ Final Answers Summary:

1. `21y² + 4y − 1`
2. `m² + 7m`
3. `3x² − 2x + 2`
4. `2r⁴ + 9r³`
5. `−6n³`
6. `18x⁵ + 6x⁴ + 12x³`
7. `−4x² + 9x − 2`
8. `5x³ − 6x + 3`
9. `−x³ + 11x² − 8x + 10`
10. `−2x³ + 12x² − 8x + 10`
11. `3x⁴ − 5x² + x − 9`
12. `14x⁴ − 4x + 3`
13. `6x⁴ − 6x`
14. `2n⁵ − 2n⁴ + 3n + 3`
15. `−14n⁴ + 7n² + 8n + 7`

Let me know if you want these written in standard form (descending powers) or need any steps explained further!