1) Greatest Common Monomial Factor
Example 1: 4x² + 8x
Solution: Factored form is 4x(x + 2)
Example 2: 12a³b − 36a²b²
Solution: Factored form is 12a²b(a − 3b)
Example 3: 5x³y⁴ − 30x²y³
Solution: Factored form is 5x²y³(x − 6)
Example 4: 9m²n³ − 12mn²
Solution: Factored form is 3mn²(3mn − 4)
Example 5: 8x⁴y² − 12x³y³
Solution: Factored form is 4x³y²(2x − 3y)
2) Difference of Two Squares
Example 1: x² − 9
Solution: Factored form is (x+3)(x−3)
Example 2: 25a² − 36
Solution: Factored form is (5a+6)(5a−6)
Example 3: 16y² − z²
Solution: Factored form is (4y+z)(4y−z)
Example 4: 49 − 4x²
Solution: Factored form is (7+2x)(7−2x)
Example 5: a² − 16b²
Solution: Factored form is (a+4b)(a−4b)
3) Difference of Two Cubes
Example 1: 8x³ − 27
Solution: Factored form is (2x−3)(4x²+6x+9)
Example 2: 64y³ − 1
Solution: Factored form is (4y−1)(16y²+4y+1)
Example 3: 27m³ − 125n³
Solution: Factored form is (3m−5n)(9m²+15mn+25n²)
Example 4: 1 − x³
Solution: Factored form is (1−x)(1+x+x²)
Example 5: 125a³ − 343b³
Solution: Factored form is (5a−7b)(25a²+35ab+49b²)
4) Sum of Two Cubes
Example 1: 8x³ + 125
Solution: Factored form is (2x+5)(4x²−10x+25)
Example 2: 1 + 64y³
Solution: Factored form is (1+4y)(16y²−8y+1)
Example 3: 27m³ + 125n³
Solution: Factored form is (3m+5n)(9m²−15mn+25n²)
Example 4: 1 + x³
Solution: Factored form is (1+x)(1−x+x²)
Example 5: 125a³ + 343b³
Solution: Factored form is (5a+7b)(25a²−35ab+49b²)
Factoring the Sum of Two Cubes (a^3 + b^3):
1) a^3 + 8
Answer: (a + 2)(a^2 - 2a + 4)
2) x^3 + 1
Answer: (x + 1)(x^2 - x + 1)
3) 27x^3 + 64
Answer: (3x + 4)(9x^2 - 12x + 16)
4) 8a^3 + 125
Answer: (2a + 5)(4a^2 - 10a + 25)
5) 1 + 64y^3
Answer: (1 + 4y)(16y^2 - 4y + 1)
Factoring the Difference of Two Cubes (a^3 - b^3):
6) x^3 – 27
Answer: (x - 3)(x^2 + 3x + 9)
7) 64m^3 – 1
Answer: (4m - 1)(16m^2 + 4m + 1)
8) 27x^3 - 64y^3
Answer: (3x - 4y)(9x^2 + 12xy + 16y^2)
9) a^3 – 125
Answer: (a - 5)(a^2 + 5a + 25)
10) 64x^3 – 1
Answer: (4x - 1)(16x^2 + 4x + 1)
Mixed (Sum and Difference) of Two Cubes:
11) x^3 + 64y^3
Answer: (x + 4y)(x^2 - 4xy + 16y^2)
12) 8a^3 – 27
Answer: (2a - 3)(4a^2 + 6a + 9)
13) 1 - 8x^3
Answer: (1 - 2x)(1 + 2x + 4x^2)
14) 125a^3 + 1
Answer: (5a + 1)(25a^2 - 5a + 1)
15) x^3 - 125y^3
Answer: (x - 5y)(x^2 + 5xy + 25y^2)
16) 1 + 27a^3
Answer: (1 + 3a)(1 - 3a + 9a^2)
17) 64 - 8x^3
Answer: 8(2 - x)(1 + x + x^2)
18) 27 - 8y^3
Answer: (3 - 2y)(9 + 6y + 4y^2)
19) x^3 + 8y^3
Answer: (x + 2y)(x^2 - 2xy + 4y^2)
20) 27x6y9 − 125z³
Answer: (3x2y3 - 5z)(9x4y6 + 15z + 25z2)