Worksheet for practicing the expansion of three-bracket algebraic expressions.

You've uploaded an image titled “Expanding Three Brackets”, which lists 15 algebraic expressions that each involve the product of three binomials (or a cube). The task is to expand each expression — that is, multiply them out completely and simplify into a polynomial in standard form.

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I’ll solve all 15 problems, showing step-by-step work for the first few, then give concise answers for the rest with brief explanations where helpful.

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## 🔢 General Strategy:

To expand three brackets like `(x+a)(x+b)(x+c)`:

1. First, multiply any two brackets using FOIL or distributive property.
2. Then multiply the resulting quadratic by the third bracket.
3. Combine like terms.

For cubes like `(x+a)³`, you can use the binomial expansion formula:
> `(x + a)³ = x³ + 3ax² + 3a²x + a³`

Or just multiply `(x+a)(x+a)(x+a)` step by step.

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# ✔ Solutions:

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1. (x + 1)(x + 2)(x + 3)

First: `(x + 1)(x + 2) = x² + 2x + x + 2 = x² + 3x + 2`

Then: `(x² + 3x + 2)(x + 3)`

= `x²(x + 3) + 3x(x + 3) + 2(x + 3)`

= `x³ + 3x² + 3x² + 9x + 2x + 6`

= `x³ + 6x² + 11x + 6`

✔ Answer: `x³ + 6x² + 11x + 6`

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2. (x + 1)(x + 2)(x + 4)

First: `(x + 1)(x + 2) = x² + 3x + 2`

Then: `(x² + 3x + 2)(x + 4)`

= `x³ + 4x² + 3x² + 12x + 2x + 8`

= `x³ + 7x² + 14x + 8`

✔ Answer: `x³ + 7x² + 14x + 8`

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3. (x + 1)(x + 2)(x + 5)

First: `(x + 1)(x + 2) = x² + 3x + 2`

Then: `(x² + 3x + 2)(x + 5)`

= `x³ + 5x² + 3x² + 15x + 2x + 10`

= `x³ + 8x² + 17x + 10`

✔ Answer: `x³ + 8x² + 17x + 10`

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4. (x + 1)(x + 2)(x + 6)

First: `(x + 1)(x + 2) = x² + 3x + 2`

Then: `(x² + 3x + 2)(x + 6)`

= `x³ + 6x² + 3x² + 18x + 2x + 12`

= `x³ + 9x² + 20x + 12`

✔ Answer: `x³ + 9x² + 20x + 12`

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5. (x + 6)(x + 1)(x + 2)

*(Same as #1, just reordered)*

Order doesn’t matter → same as #1

✔ Answer: `x³ + 6x² + 11x + 6`

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6. (x + 6)(x + 1)(x - 2)

First: `(x + 6)(x + 1) = x² + 7x + 6`

Then: `(x² + 7x + 6)(x - 2)`

= `x³ - 2x² + 7x² - 14x + 6x - 12`

= `x³ + 5x² - 8x - 12`

✔ Answer: `x³ + 5x² - 8x - 12`

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7. (x + 6)(x + 1)(x - 3)

First: `(x + 6)(x + 1) = x² + 7x + 6`

Then: `(x² + 7x + 6)(x - 3)`

= `x³ - 3x² + 7x² - 21x + 6x - 18`

= `x³ + 4x² - 15x - 18`

✔ Answer: `x³ + 4x² - 15x - 18`

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8. (x + 6)(x + 3)(x - 3)

Notice: `(x + 3)(x - 3) = x² - 9` → difference of squares!

So: `(x + 6)(x² - 9)`

= `x³ - 9x + 6x² - 54`

= `x³ + 6x² - 9x - 54`

✔ Answer: `x³ + 6x² - 9x - 54`

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9. (x + 6)(x - 3)(x - 3)

→ `(x + 6)(x - 3)²`

First: `(x - 3)² = x² - 6x + 9`

Then: `(x + 6)(x² - 6x + 9)`

= `x³ - 6x² + 9x + 6x² - 36x + 54`

= `x³ - 27x + 54`

✔ Answer: `x³ - 27x + 54`

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10. (x + 6)(x - 4)(x - 4)

→ `(x + 6)(x - 4)²`

`(x - 4)² = x² - 8x + 16`

Multiply: `(x + 6)(x² - 8x + 16)`

= `x³ - 8x² + 16x + 6x² - 48x + 96`

= `x³ - 2x² - 32x + 96`

✔ Answer: `x³ - 2x² - 32x + 96`

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11. (x + 4)(x - 4)(x - 4)

→ `(x + 4)(x - 4)²`

First: `(x - 4)² = x² - 8x + 16`

Then: `(x + 4)(x² - 8x + 16)`

= `x³ - 8x² + 16x + 4x² - 32x + 64`

= `x³ - 4x² - 16x + 64`

✔ Answer: `x³ - 4x² - 16x + 64`

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12. (x + 4)(x - 4)²

*(Same as #11!)*

✔ Answer: `x³ - 4x² - 16x + 64`

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13. (x + 4)³

Use binomial expansion:

= `x³ + 3·x²·4 + 3·x·4² + 4³`

= `x³ + 12x² + 48x + 64`

✔ Answer: `x³ + 12x² + 48x + 64`

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14. (x - 4)³

= `x³ - 3·x²·4 + 3·x·16 - 64`

= `x³ - 12x² + 48x - 64`

✔ Answer: `x³ - 12x² + 48x - 64`

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15. (3x - 4)³

Use binomial expansion:

= `(3x)³ - 3·(3x)²·4 + 3·(3x)·4² - 4³`

= `27x³ - 3·9x²·4 + 3·3x·16 - 64`

= `27x³ - 108x² + 144x - 64`

✔ Answer: `27x³ - 108x² + 144x - 64`

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## 📋 Final Answer Summary:

Here are all 15 expanded forms:

1. `x³ + 6x² + 11x + 6`
2. `x³ + 7x² + 14x + 8`
3. `x³ + 8x² + 17x + 10`
4. `x³ + 9x² + 20x + 12`
5. `x³ + 6x² + 11x + 6`
6. `x³ + 5x² - 8x - 12`
7. `x³ + 4x² - 15x - 18`
8. `x³ + 6x² - 9x - 54`
9. `x³ - 27x + 54`
10. `x³ - 2x² - 32x + 96`
11. `x³ - 4x² - 16x + 64`
12. `x³ - 4x² - 16x + 64`
13. `x³ + 12x² + 48x + 64`
14. `x³ - 12x² + 48x - 64`
15. `27x³ - 108x² + 144x - 64`

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Let me know if you’d like to see factoring or graphing of these polynomials next! 😊