MATH LESSON
CUBE OF A BINOMIAL
Learn the shortcut to expanding the cube of a binomial without doing long multiplication!
"Cube of a binomial results in a 4-term polynomial where the coefficients of the middle terms follow the pattern 1, 3, 3, 1."
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
Understanding the Pattern
When you cube a binomial, the result always has 4 terms. The exponents of 'a' decrease (3, 2, 1, 0) while the exponents of 'b' increase (0, 1, 2, 3). The coefficients are always 1, 3, 3, 1.
Term 1: a³ — coefficient is 1
Term 2: 3a²b — coefficient is 3
Term 3: 3ab² — coefficient is 3
Term 4: b³ — coefficient is 1
Watch the Signs!
For (a + b)³, ALL terms are POSITIVE.
For (a − b)³, the signs ALTERNATE (+, −, +, −).
Worked Examples (Sum)
Example 1: (x + 1)³
a=x, b=1 | Pattern: 1, 3, 3, 1
→ x³ + 3x² + 3x + 1
a=x, b=1 | Pattern: 1, 3, 3, 1
→ x³ + 3x² + 3x + 1
Example 2: (x + 2)³
a=x, b=2 | 1(x³) + 3(x²)(2) + 3(x)(4) + 8
→ x³ + 6x² + 12x + 8
a=x, b=2 | 1(x³) + 3(x²)(2) + 3(x)(4) + 8
→ x³ + 6x² + 12x + 8
Example 3: (2x + 3)³
a=2x, b=3 | 1(8x³) + 3(4x²)(3) + 3(2x)(9) + 27
→ 8x³ + 36x² + 54x + 27
a=2x, b=3 | 1(8x³) + 3(4x²)(3) + 3(2x)(9) + 27
→ 8x³ + 36x² + 54x + 27
Example 4: (x + 4)³
a=x, b=4 | 1(x³) + 3(x²)(4) + 3(x)(16) + 64
→ x³ + 12x² + 48x + 64
a=x, b=4 | 1(x³) + 3(x²)(4) + 3(x)(16) + 64
→ x³ + 12x² + 48x + 64
Example 5: (x + 5)³
a=x, b=5 | 1(x³) + 3(x²)(5) + 3(x)(25) + 125
→ x³ + 15x² + 75x + 125
a=x, b=5 | 1(x³) + 3(x²)(5) + 3(x)(25) + 125
→ x³ + 15x² + 75x + 125
Example 6: (x + 6)³
a=x, b=6 | 1(x³) + 3(x²)(6) + 3(x)(36) + 216
→ x³ + 18x² + 108x + 216
a=x, b=6 | 1(x³) + 3(x²)(6) + 3(x)(36) + 216
→ x³ + 18x² + 108x + 216
Example 7: (2x + 5)³
a=2x, b=5 | 1(8x³) + 3(4x²)(5) + 3(2x)(25) + 125
→ 8x³ + 60x² + 150x + 125
a=2x, b=5 | 1(8x³) + 3(4x²)(5) + 3(2x)(25) + 125
→ 8x³ + 60x² + 150x + 125
Worked Examples (Difference)
Example 8: (x − 1)³
a=x, b=1 (alternate signs)
→ x³ − 3x² + 3x − 1
a=x, b=1 (alternate signs)
→ x³ − 3x² + 3x − 1
Example 9: (x − 2)³
a=x, b=2 | 1(x³) − 3(x²)(2) + 3(x)(4) − 8
→ x³ − 6x² + 12x − 8
a=x, b=2 | 1(x³) − 3(x²)(2) + 3(x)(4) − 8
→ x³ − 6x² + 12x − 8
Example 10: (x − 3)³
a=x, b=3 | 1(x³) − 3(x²)(3) + 3(x)(9) − 27
→ x³ − 9x² + 27x − 27
a=x, b=3 | 1(x³) − 3(x²)(3) + 3(x)(9) − 27
→ x³ − 9x² + 27x − 27
Example 11: (x − 5)³
a=x, b=5 | 1(x³) − 3(x²)(5) + 3(x)(25) − 125
→ x³ − 15x² + 75x − 125
a=x, b=5 | 1(x³) − 3(x²)(5) + 3(x)(25) − 125
→ x³ − 15x² + 75x − 125
Example 12: (x − 6)³
a=x, b=6 | 1(x³) − 3(x²)(6) + 3(x)(36) − 216
→ x³ − 18x² + 108x − 216
a=x, b=6 | 1(x³) − 3(x²)(6) + 3(x)(36) − 216
→ x³ − 18x² + 108x − 216
Example 13: (3x − 1)³
a=3x, b=1 | 1(27x³) − 3(9x²)(1) + 3(3x)(1) − 1
→ 27x³ − 27x² + 9x − 1
a=3x, b=1 | 1(27x³) − 3(9x²)(1) + 3(3x)(1) − 1
→ 27x³ − 27x² + 9x − 1
Example 14: (2x − 4)³
a=2x, b=4 | 1(8x³) − 3(4x²)(4) + 3(2x)(16) − 64
→ 8x³ − 48x² + 96x − 64
a=2x, b=4 | 1(8x³) − 3(4x²)(4) + 3(2x)(16) − 64
→ 8x³ − 48x² + 96x − 64
Shortcut: Just remember the coefficient pattern 1, 3, 3, 1 and alternate signs if it's a difference!
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