Factoring Practice worksheet featuring 21 algebraic expressions for students to factor, including quadratic, difference of squares, and trigonometric expressions.

You're looking at a Factoring Practice worksheet that uses unconventional symbols (like `#`, `@`, `∂`, `%`, `$`, `♦`, `θ`, etc.) instead of standard variables like `x` or `y`. This is likely done to help students focus on the structure of the expressions rather than getting distracted by familiar variables.

Let’s solve each problem one by one, explaining the factoring method used.

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1. `#² - !²`

This is a difference of squares:
> `a² - b² = (a - b)(a + b)`

So:
✔ Answer: `(# - !)(# + !)`

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2. `#² - 3# - 28`

This is a quadratic trinomial. We look for two numbers that:
- Multiply to `-28`
- Add to `-3`

Those numbers are `-7` and `+4`.

✔ Answer: `(# - 7)(# + 4)`

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3. `@² + 6@ + 9`

This is a perfect square trinomial:
> `a² + 2ab + b² = (a + b)²`

Here, `@² + 6@ + 9 = (@ + 3)²`

✔ Answer: `(@ + 3)²`

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4. `3∂² + 8∂ + 4`

Factor this quadratic. Use AC method or trial:

We want two numbers that multiply to `3×4 = 12` and add to `8` → `6` and `2`

Split middle term:
`3∂² + 6∂ + 2∂ + 4`
Group: `(3∂² + 6∂) + (2∂ + 4)`
Factor: `3∂(∂ + 2) + 2(∂ + 2)`
→ `(3∂ + 2)(∂ + 2)`

✔ Answer: `(3∂ + 2)(∂ + 2)`

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5. `(F(x))² - 2(F(x)) - 8`

Treat `F(x)` as a single variable, say `u`.

Then: `u² - 2u - 8` → factor: `(u - 4)(u + 2)`

Replace back:
✔ Answer: `(F(x) - 4)(F(x) + 2)`

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6. `(sin x)² - (cos x)²`

This is again a difference of squares:
> `a² - b² = (a - b)(a + b)`

So:
✔ Answer: `(sin x - cos x)(sin x + cos x)`

*(Note: This is also equal to `-cos(2x)` using trig identity, but since it's factoring practice, we leave it factored.)*

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16. `%² - $²`

Again, difference of squares.

✔ Answer: `(% - $)(% + $)`

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17. `♦³ - θ³`

This is a difference of cubes:
> `a³ - b³ = (a - b)(a² + ab + b²)`

✔ Answer: `(♦ - θ)(♦² + ♦θ + θ²)`

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18. `(y - 1)² - (x + 2)²`

Difference of squares! Let `a = y - 1`, `b = x + 2`

Then: `(a - b)(a + b)` → `[(y - 1) - (x + 2)][(y - 1) + (x + 2)]`

Simplify:
First factor: `y - 1 - x - 2 = y - x - 3`
Second factor: `y - 1 + x + 2 = y + x + 1`

✔ Answer: `(y - x - 3)(y + x + 1)`

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19. `(x - 3)² - 9`

Note: `9 = 3²`, so this is again difference of squares.

Let `a = x - 3`, `b = 3`

→ `(a - b)(a + b) = (x - 3 - 3)(x - 3 + 3) = (x - 6)(x)`

✔ Answer: `x(x - 6)`

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20. `(3x + 9)² - 81`

Again, difference of squares:
`81 = 9²`, so:

`[(3x + 9) - 9][(3x + 9) + 9]`

Simplify:
First: `3x + 9 - 9 = 3x`
Second: `3x + 9 + 9 = 3x + 18`

So: `3x(3x + 18)` → you can factor out 3 from second term: `3x * 3(x + 6) = 9x(x + 6)`

✔ Answer: `9x(x + 6)`

*(Alternatively, leave as `3x(3x + 18)` — both are correct, but fully factored is better.)*

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21. `$² - 1`

Difference of squares: `a² - 1² = (a - 1)(a + 1)`

✔ Answer: `($ - 1)($ + 1)`

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## ✔ Final Answers Summary:

1. `(# - !)(# + !)`
2. `(# - 7)(# + 4)`
3. `(@ + 3)²`
4. `(3∂ + 2)(∂ + 2)`
5. `(F(x) - 4)(F(x) + 2)`
6. `(sin x - cos x)(sin x + cos x)`
16. `(% - $)(% + $)`
17. `(♦ - θ)(♦² + ♦θ + θ²)`
18. `(y - x - 3)(y + x + 1)`
19. `x(x - 6)`
20. `9x(x + 6)`
21. `($ - 1)($ + 1)`

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💡 Key Takeaway: Even though the symbols are strange, the factoring patterns are standard:
- Difference of squares
- Perfect square trinomials
- Factoring quadratics
- Difference of cubes
- Substitution (treating functions or expressions as variables)

This worksheet is excellent for practicing recognizing structure over memorizing variables!

Let me know if you’d like to see step-by-step work for any specific problem!