Factoring sum and difference of two cubes practice worksheet with worked examples.

It looks like you've uploaded an image of a worksheet titled "Polynomials - Factoring Sum and Difference of Two Cubes". The worksheet contains several polynomial expressions, each labeled with a code (e.g., THE, THD, TOLD, etc.), and the goal is to factor them using the sum or difference of two cubes formulas.

Let me explain the method, then go through a few examples from the worksheet to demonstrate how to solve them.

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🔷 Factoring Sum and Difference of Two Cubes

There are two key formulas:

1. Sum of cubes:
$$
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
$$

2. Difference of cubes:
$$
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
$$

We need to identify if the given expression fits one of these forms.

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Let’s go through some examples from your worksheet.

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✔ Example 1: THE — $8x^3 + 125$

This is a sum of cubes:
- $8x^3 = (2x)^3$
- $125 = 5^3$

So,
$$
8x^3 + 125 = (2x)^3 + (5)^3 = (2x + 5)((2x)^2 - (2x)(5) + 5^2)
$$
$$
= (2x + 5)(4x^2 - 10x + 25)
$$

✔ Answer: $(2x + 5)(4x^2 - 10x + 25)$

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✔ Example 2: THD — $27x^3 - 64y^3$

This is a difference of cubes:
- $27x^3 = (3x)^3$
- $64y^3 = (4y)^3$

So,
$$
27x^3 - 64y^3 = (3x)^3 - (4y)^3 = (3x - 4y)((3x)^2 + (3x)(4y) + (4y)^2)
$$
$$
= (3x - 4y)(9x^2 + 12xy + 16y^2)
$$

✔ Answer: $(3x - 4y)(9x^2 + 12xy + 16y^2)$

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✔ Example 3: TOLD — $x^3 - 8$

This is a difference of cubes:
- $x^3 = x^3$
- $8 = 2^3$

So,
$$
x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)
$$

✔ Answer: $(x - 2)(x^2 + 2x + 4)$

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✔ Example 4: BOYS — $27x^3 + 64$

This is a sum of cubes:
- $27x^3 = (3x)^3$
- $64 = 4^3$

So,
$$
27x^3 + 64 = (3x)^3 + 4^3 = (3x + 4)((3x)^2 - (3x)(4) + 4^2)
$$
$$
= (3x + 4)(9x^2 - 12x + 16)
$$

✔ Answer: $(3x + 4)(9x^2 - 12x + 16)$

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✔ Example 5: THE — $64x^3 - 1$ (second row)

This is a difference of cubes:
- $64x^3 = (4x)^3$
- $1 = 1^3$

So,
$$
64x^3 - 1 = (4x)^3 - 1^3 = (4x - 1)((4x)^2 + (4x)(1) + 1^2)
$$
$$
= (4x - 1)(16x^2 + 4x + 1)
$$

✔ Answer: $(4x - 1)(16x^2 + 4x + 1)$

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✔ Example 6: THD — $216x^3 - 343$

Note: $216 = 6^3$, $343 = 7^3$, so this is a difference of cubes:
- $216x^3 = (6x)^3$
- $343 = 7^3$

So,
$$
(6x)^3 - 7^3 = (6x - 7)((6x)^2 + (6x)(7) + 7^2)
= (6x - 7)(36x^2 + 42x + 49)
$$

✔ Answer: $(6x - 7)(36x^2 + 42x + 49)$

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✔ Example 7: TOLD — $125x^3 + 1$

This is a sum of cubes:
- $125x^3 = (5x)^3$
- $1 = 1^3$

So,
$$
(5x)^3 + 1^3 = (5x + 1)((5x)^2 - (5x)(1) + 1^2)
= (5x + 1)(25x^2 - 5x + 1)
$$

✔ Answer: $(5x + 1)(25x^2 - 5x + 1)$

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✔ Example 8: BOYS — $x^3 - 64$

This is a difference of cubes:
- $x^3 = x^3$
- $64 = 4^3$

So,
$$
x^3 - 64 = x^3 - 4^3 = (x - 4)(x^2 + 4x + 16)
$$

✔ Answer: $(x - 4)(x^2 + 4x + 16)$

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✔ Example 9: THE — $125x^3 - 1$ (bottom left)

This is a difference of cubes:
- $125x^3 = (5x)^3$
- $1 = 1^3$

So,
$$
(5x)^3 - 1^3 = (5x - 1)((5x)^2 + (5x)(1) + 1^2)
= (5x - 1)(25x^2 + 5x + 1)
$$

✔ Answer: $(5x - 1)(25x^2 + 5x + 1)$

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✔ Example 10: THD — $27x^3 - 1$

This is a difference of cubes:
- $27x^3 = (3x)^3$
- $1 = 1^3$

So,
$$
(3x)^3 - 1^3 = (3x - 1)((3x)^2 + (3x)(1) + 1^2)
= (3x - 1)(9x^2 + 3x + 1)
$$

✔ Answer: $(3x - 1)(9x^2 + 3x + 1)$

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✔ Example 11: TOLD — $27x^3 + 1$

This is a sum of cubes:
- $27x^3 = (3x)^3$
- $1 = 1^3$

So,
$$
(3x)^3 + 1^3 = (3x + 1)((3x)^2 - (3x)(1) + 1^2)
= (3x + 1)(9x^2 - 3x + 1)
$$

✔ Answer: $(3x + 1)(9x^2 - 3x + 1)$

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✔ Example 12: BOYS — $64x^3 + 27$

This is a sum of cubes:
- $64x^3 = (4x)^3$
- $27 = 3^3$

So,
$$
(4x)^3 + 3^3 = (4x + 3)((4x)^2 - (4x)(3) + 3^2)
= (4x + 3)(16x^2 - 12x + 9)
$$

✔ Answer: $(4x + 3)(16x^2 - 12x + 9)$

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## 🧠 Summary of Strategy

To factor sum/difference of cubes:

1. Check if it's a sum or difference of two terms.
2. See if both terms are perfect cubes.
3. Apply the appropriate formula:
- Sum: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
- Diff: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
4. Simplify each part carefully.

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If you want, I can provide a complete table of all answers from the worksheet in the same format as the original.

Let me know if you'd like that!