Factoring The Sum Or Difference Of Cubes Color By Number | And ... - Free Printable

Problem Overview:

The task involves factoring each polynomial completely using the sum/difference of cubes formula. After factoring, we calculate the sum of the coefficients of each term in the factored polynomial and use that number to color a corresponding section on a picture (not shown here). Let's solve each polynomial step by step.

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

Step-by-Step Solutions:

#### 1. \(x^3 + 125\)
- Recognize: \(125 = 5^3\), so this is a sum of cubes.
- Apply the sum of cubes formula:
\[
x^3 + 125 = (x + 5)(x^2 - 5x + 25)
\]
- Factored Polynomial: \((x + 5)(x^2 - 5x + 25)\)
- Sum of Coefficients:
\[
(1 + 5) + (1 - 5 + 25) = 6 + 21 = 27
\]
- Color: Red

#### 2. \(x^3 + 64\)
- Recognize: \(64 = 4^3\), so this is a sum of cubes.
- Apply the sum of cubes formula:
\[
x^3 + 64 = (x + 4)(x^2 - 4x + 16)
\]
- Factored Polynomial: \((x + 4)(x^2 - 4x + 16)\)
- Sum of Coefficients:
\[
(1 + 4) + (1 - 4 + 16) = 5 + 13 = 18
\]
- Color: Light Blue

#### 3. \(x^3 - 8\)
- Recognize: \(8 = 2^3\), so this is a difference of cubes.
- Apply the difference of cubes formula:
\[
x^3 - 8 = (x - 2)(x^2 + 2x + 4)
\]
- Factored Polynomial: \((x - 2)(x^2 + 2x + 4)\)
- Sum of Coefficients:
\[
(1 - 2) + (1 + 2 + 4) = -1 + 7 = 6
\]
- Color: Dark Blue

#### 4. \(x^3 - 729\)
- Recognize: \(729 = 9^3\), so this is a difference of cubes.
- Apply the difference of cubes formula:
\[
x^3 - 729 = (x - 9)(x^2 + 9x + 81)
\]
- Factored Polynomial: \((x - 9)(x^2 + 9x + 81)\)
- Sum of Coefficients:
\[
(1 - 9) + (1 + 9 + 81) = -8 + 91 = 83
\]
- Color: Purple

#### 5. \(1 - x^3\)
- Recognize: This is a difference of cubes, but rewrite it as \(-x^3 + 1\).
- Apply the difference of cubes formula:
\[
1 - x^3 = (1 - x)(1 + x + x^2)
\]
- Factored Polynomial: \((1 - x)(1 + x + x^2)\)
- Sum of Coefficients:
\[
(1 - 1) + (1 + 1 + 1) = 0 + 3 = 3
\]
- Color: Pink

#### 6. \(x^3 - 343\)
- Recognize: \(343 = 7^3\), so this is a difference of cubes.
- Apply the difference of cubes formula:
\[
x^3 - 343 = (x - 7)(x^2 + 7x + 49)
\]
- Factored Polynomial: \((x - 7)(x^2 + 7x + 49)\)
- Sum of Coefficients:
\[
(1 - 7) + (1 + 7 + 49) = -6 + 57 = 51
\]
- Color: Yellow

#### 7. \(8x^3 + 27\)
- Recognize: \(8x^3 = (2x)^3\) and \(27 = 3^3\), so this is a sum of cubes.
- Apply the sum of cubes formula:
\[
8x^3 + 27 = (2x + 3)((2x)^2 - (2x)(3) + 3^2) = (2x + 3)(4x^2 - 6x + 9)
\]
- Factored Polynomial: \((2x + 3)(4x^2 - 6x + 9)\)
- Sum of Coefficients:
\[
(2 + 3) + (4 - 6 + 9) = 5 + 7 = 12
\]
- Color: Red

#### 8. \(-x^3 - 8\)
- Recognize: This can be rewritten as \(-(x^3 + 8)\), where \(x^3 + 8\) is a sum of cubes.
- Factor \(x^3 + 8\) first:
\[
x^3 + 8 = (x + 2)(x^2 - 2x + 4)
\]
- Then:
\[
-x^3 - 8 = -(x + 2)(x^2 - 2x + 4)
\]
- Factored Polynomial: \(-(x + 2)(x^2 - 2x + 4)\)
- Sum of Coefficients:
\[
-(1 + 2) + -(1 - 2 + 4) = -3 + (-3) = -6
\]
- Color: Light Blue

#### 9. \(512x^3 - 729\)
- Recognize: \(512x^3 = (8x)^3\) and \(729 = 9^3\), so this is a difference of cubes.
- Apply the difference of cubes formula:
\[
512x^3 - 729 = (8x - 9)((8x)^2 + (8x)(9) + 9^2) = (8x - 9)(64x^2 + 72x + 81)
\]
- Factored Polynomial: \((8x - 9)(64x^2 + 72x + 81)\)
- Sum of Coefficients:
\[
(8 - 9) + (64 + 72 + 81) = -1 + 217 = 216
\]
- Color: Dark Blue

#### 10. \(8x^3 + 125\)
- Recognize: \(8x^3 = (2x)^3\) and \(125 = 5^3\), so this is a sum of cubes.
- Apply the sum of cubes formula:
\[
8x^3 + 125 = (2x + 5)((2x)^2 - (2x)(5) + 5^2) = (2x + 5)(4x^2 - 10x + 25)
\]
- Factored Polynomial: \((2x + 5)(4x^2 - 10x + 25)\)
- Sum of Coefficients:
\[
(2 + 5) + (4 - 10 + 25) = 7 + 19 = 26
\]
- Color: Purple

#### 11. \(8x^4 + x\)
- Factor out the greatest common factor (GCF):
\[
8x^4 + x = x(8x^3 + 1)
\]
- Recognize: \(8x^3 + 1\) is a sum of cubes.
- Apply the sum of cubes formula:
\[
8x^3 + 1 = (2x + 1)((2x)^2 - (2x)(1) + 1^2) = (2x + 1)(4x^2 - 2x + 1)
\]
- Factored Polynomial: \(x(2x + 1)(4x^2 - 2x + 1)\)
- Sum of Coefficients:
\[
(1) + (2 + 1) + (4 - 2 + 1) = 1 + 3 + 3 = 7
\]
- Color: Pink

#### 12. \(125x^3y + 27y\)
- Factor out the GCF:
\[
125x^3y + 27y = y(125x^3 + 27)
\]
- Recognize: \(125x^3 = (5x)^3\) and \(27 = 3^3\), so \(125x^3 + 27\) is a sum of cubes.
- Apply the sum of cubes formula:
\[
125x^3 + 27 = (5x + 3)((5x)^2 - (5x)(3) + 3^2) = (5x + 3)(25x^2 - 15x + 9)
\]
- Factored Polynomial: \(y(5x + 3)(25x^2 - 15x + 9)\)
- Sum of Coefficients:
\[
(1) + (5 + 3) + (25 - 15 + 9) = 1 + 8 + 19 = 28
\]
- Color: Yellow

Final Answer:

\[
\boxed{
\begin{array}{|c|c|c|c|}
\hline
\text{Polynomial} & \text{Factored Polynomial} & \text{Sum of Coefficients} & \text{Color} \\
\hline
1. \ x^3 + 125 & (x + 5)(x^2 - 5x + 25) & 27 & \text{Red} \\
2. \ x^3 + 64 & (x + 4)(x^2 - 4x + 16) & 18 & \text{Light Blue} \\
3. \ x^3 - 8 & (x - 2)(x^2 + 2x + 4) & 6 & \text{Dark Blue} \\
4. \ x^3 - 729 & (x - 9)(x^2 + 9x + 81) & 83 & \text{Purple} \\
5. \ 1 - x^3 & (1 - x)(1 + x + x^2) & 3 & \text{Pink} \\
6. \ x^3 - 343 & (x - 7)(x^2 + 7x + 49) & 51 & \text{Yellow} \\
7. \ 8x^3 + 27 & (2x + 3)(4x^2 - 6x + 9) & 12 & \text{Red} \\
8. \ -x^3 - 8 & -(x + 2)(x^2 - 2x + 4) & -6 & \text{Light Blue} \\
9. \ 512x^3 - 729 & (8x - 9)(64x^2 + 72x + 81) & 216 & \text{Dark Blue} \\
10. \ 8x^3 + 125 & (2x + 5)(4x^2 - 10x + 25) & 26 & \text{Purple} \\
11. \ 8x^4 + x & x(2x + 1)(4x^2 - 2x + 1) & 7 & \text{Pink} \\
12. \ 125x^3y + 27y & y(5x + 3)(25x^2 - 15x + 9) & 28 & \text{Yellow} \\
\hline
\end{array}
}
\]