Fun Math Activity: Color by Number - Factoring the Sum or Difference of Cubes.

Let’s solve this step by step.

We’re being asked to factor expressions that are either the sum of cubes or the difference of cubes.

There are two special formulas you need to remember:

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

2. Difference of Cubes:
\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)

These formulas work for any numbers or variables — as long as you can write the expression as something cubed plus or minus something else cubed.

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Let’s look at each problem one by one and apply the right formula.

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Problem 51: \( x^3 + 8 \)

This is a sum of cubes because:
- \( x^3 \) is already a cube.
- \( 8 = 2^3 \), so it’s also a cube.

So, \( a = x \), \( b = 2 \)

Use sum of cubes formula:
\( (x + 2)(x^2 - 2x + 4) \)

✔ Answer for 51: \( (x + 2)(x^2 - 2x + 4) \)

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Problem 2: \( y^3 - 27 \)

This is a difference of cubes:
- \( y^3 \) is a cube.
- \( 27 = 3^3 \)

So, \( a = y \), \( b = 3 \)

Use difference of cubes:
\( (y - 3)(y^2 + 3y + 9) \)

✔ Answer for 2: \( (y - 3)(y^2 + 3y + 9) \)

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Problem 0: \( 64z^3 - 1 \)

Difference of cubes:
- \( 64z^3 = (4z)^3 \)
- \( 1 = 1^3 \)

So, \( a = 4z \), \( b = 1 \)

Formula: \( (4z - 1)((4z)^2 + (4z)(1) + 1^2) \)

Simplify:
\( (4z - 1)(16z^2 + 4z + 1) \)

✔ Answer for 0: \( (4z - 1)(16z^2 + 4z + 1) \)

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Problem 83: \( 125m^3 + n^3 \)

Sum of cubes:
- \( 125m^3 = (5m)^3 \)
- \( n^3 = n^3 \)

So, \( a = 5m \), \( b = n \)

Formula: \( (5m + n)((5m)^2 - (5m)(n) + n^2) \)

Simplify:
\( (5m + n)(25m^2 - 5mn + n^2) \)

✔ Answer for 83: \( (5m + n)(25m^2 - 5mn + n^2) \)

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Problem 21: \( p^3 + q^3 \)

Straightforward sum of cubes:
\( a = p \), \( b = q \)

Formula: \( (p + q)(p^2 - pq + q^2) \)

✔ Answer for 21: \( (p + q)(p^2 - pq + q^2) \)

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Problem 18: \( r^3 - s^3 \)

Difference of cubes:
\( a = r \), \( b = s \)

Formula: \( (r - s)(r^2 + rs + s^2) \)

✔ Answer for 18: \( (r - s)(r^2 + rs + s^2) \)

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Problem 3: \( 8t^3 + 27u^3 \)

Sum of cubes:
- \( 8t^3 = (2t)^3 \)
- \( 27u^3 = (3u)^3 \)

So, \( a = 2t \), \( b = 3u \)

Formula: \( (2t + 3u)((2t)^2 - (2t)(3u) + (3u)^2) \)

Simplify:
\( (2t + 3u)(4t^2 - 6tu + 9u^2) \)

✔ Answer for 3: \( (2t + 3u)(4t^2 - 6tu + 9u^2) \)

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Problem 12: \( v^3 - 64w^3 \)

Difference of cubes:
- \( v^3 = v^3 \)
- \( 64w^3 = (4w)^3 \)

So, \( a = v \), \( b = 4w \)

Formula: \( (v - 4w)(v^2 + v(4w) + (4w)^2) \)

Simplify:
\( (v - 4w)(v^2 + 4vw + 16w^2) \)

✔ Answer for 12: \( (v - 4w)(v^2 + 4vw + 16w^2) \)

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Problem 26: \( 216x^3 - y^3 \)

Difference of cubes:
- \( 216x^3 = (6x)^3 \)
- \( y^3 = y^3 \)

So, \( a = 6x \), \( b = y \)

Formula: \( (6x - y)((6x)^2 + (6x)(y) + y^2) \)

Simplify:
\( (6x - y)(36x^2 + 6xy + y^2) \)

✔ Answer for 26: \( (6x - y)(36x^2 + 6xy + y^2) \)

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Problem 7: \( z^3 + 1 \)

Sum of cubes:
- \( z^3 = z^3 \)
- \( 1 = 1^3 \)

So, \( a = z \), \( b = 1 \)

Formula: \( (z + 1)(z^2 - z + 1) \)

✔ Answer for 7: \( (z + 1)(z^2 - z + 1) \)

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Now let’s double-check a few to make sure we didn’t mix up signs.

For example, Problem 51: \( x^3 + 8 \) → sum → first factor has +, second factor has – in middle term → correct.

Problem 2: \( y^3 - 27 \) → difference → first factor has –, second factor has + in middle term → correct.

All others follow same pattern.

Also, note that sometimes the answer might be written with terms reordered, but as long as the factors match the formula, it’s fine.

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Final Answers (matching the numbers from the image):

51: \( (x + 2)(x^2 - 2x + 4) \)
2: \( (y - 3)(y^2 + 3y + 9) \)
0: \( (4z - 1)(16z^2 + 4z + 1) \)
83: \( (5m + n)(25m^2 - 5mn + n^2) \)
21: \( (p + q)(p^2 - pq + q^2) \)
18: \( (r - s)(r^2 + rs + s^2) \)
3: \( (2t + 3u)(4t^2 - 6tu + 9u^2) \)
12: \( (v - 4w)(v^2 + 4vw + 16w^2) \)
26: \( (6x - y)(36x^2 + 6xy + y^2) \)
7: \( (z + 1)(z^2 - z + 1) \)

You can now use these answers to color your mandala according to the key!

Final Answer:
51: (x + 2)(x² - 2x + 4)
2: (y - 3)(y² + 3y + 9)
0: (4z - 1)(16z² + 4z + 1)
83: (5m + n)(25m² - 5mn + n²)
21: (p + q)(p² - pq + q²)
18: (r - s)(r² + rs + s²)
3: (2t + 3u)(4t² - 6tu + 9u²)
12: (v - 4w)(v² + 4vw + 16w²)
26: (6x - y)(36x² + 6xy + y²)
7: (z + 1)(z² - z + 1)