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Difference of Cubes
After any GCF is factored out, an expression is considered a difference of cubes provided:
- Only has 2 terms
- Minus sign between the terms
- Variables have exponents that are multiples of 3, such as 𝑥
3
6
9
12
- Numbers are perfect cubes, such as 1 , 8 , 27 , 64 , 125 , 216 , 343 , 512 , …
If all 4 above are true after any GCF is factored out, then we factor as follows:
Step 1: Factor out GCF, if applicable
Step 2: Use the rule: 𝑎
3
3
2
2
Step 3: Determine “a”.
- Cube root any numbers in the first term. (if applicable)
- Divide the exponent of the first term’s variable by 3. (if applicable)
Step 4: Determine “b”
- Cube root any numbers in the first term. (if applicable)
- Divide the exponent of the first term’s variable by 3. (if applicable
Step 5: Place the numbers in the formula and simplify to get the answer.
Step 6: Check
Sum of Cubes
After any GCF is factored out, an expression is considered a sum of cubes provided:
- Only has 2 terms
- Plus sign between the terms
- Variables have exponents that are multiples of 3, such as 𝑥
3
6
9
12
- Numbers are perfect cubes, such as 1 , 8 , 27 , 64 , 125 , 216 , 343 , 512 , …
If all 4 above are true after any GCF is factored out, then we factor as follows:
Step 1: Factor out GCF, if applicable
Step 2: Create template: 𝑎
3
3
2
2
Step 3: Determine “a”.
- Cube root any numbers in the first term. (if applicable)
- Divide the exponent of the first term’s variable by 3. (if applicable)
Step 4: Determine “b”
- Cube root any numbers in the first term. (if applicable)
- Divide the exponent of the first term’s variable by 3. (if applicable
Step 5: Place the numbers in the formula and simplify to get the answer.
Step 6: Check
For Example: Factor 125𝑥
3
6
Step 1: Factor out GCF, if applicable
Step 2: Create Template:
3
3
2
2
Step 3: Determine “a”.
the first term. (if
applicable)
the first term’s variable
by 3. (if applicable)
Step 4: Determine “b”
the first term. (if
applicable)
the first term’s variable
by 3. (if applicable
Step 5: Place the numbers in the
formula and simplify to get the
answer.
Step 6: Check
Solution: There is no GCF.
Check to see if it is a sum of cubes:
- Only has 2 terms ✓
- Plus sign between the terms ✓
- Variables have exponents that are multiples of
3
6
exponent of 6 is a multiple of 3. ✓
- Numbers are perfect cubes, 125= 5
3
Problem is a sum of cubes.
Step 1: There is no GCF, skip to step 2.
Step 2: Create Template: (𝑎 + 𝑏)(𝑎
2
2
Step 3: Determine “a”. 𝑎 = √ 125
3
3 ⁄ 3
Step 3: Determine “b”. 𝑏 = 𝑦
6 ⁄ 3
2
b = 𝑦
2
Step 5: Place the numbers in the formula and simplify
to get the answer.
3
6
2
2
2
2
2
= Answer: ( 5 𝑥 + 𝑦
2
2
2
4
Step 6: Check
2
2
4
× 5 𝑥 + 𝑦
2
2
2
4
6
3
2
2
4
___
3
6
#1- 42 : Completely factor the binomials, remember to factor out the GCF first when applicable
(if a problem is prime say so).
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
- x
9
3
6
3
6
6
- 125x
9
6
9
6
3
3
3
3
4
4
4
4
5
2
5
2
3
3
3
3
6
9
