Example of Binomial Cubed

In algebra, a binomial is an expression of two terms, which are added with positive or negative signs. When binomials are multiplied, one of the so-called Remarkable products:

• Binomial squared: (a + b)², which is the same as (a + b) * (a + b)
• Conjugated binomials:(a + b) * (a - b)
• Binomials with common term:(a + b) * (a + c)
• Binomial cubed: (a + b)³, which is the same as (a + b) * (a + b) * (a + b)

This time we will talk about binomial cubed. This remarkable product is the product of the binomial itself, and again: (a + b) * (a + b) * (a + b). It is the same as raising the binomial to the exponent 3. To obtain the result of this algebraic operation, an already established rule is followed, which says:

• First term cube: (a)³ = to³
• Plus the triple product of the square of the first by the second: + 3 * (a)²* (b) = +3rd²b
• Plus the triple product of the first by the square of the second: + 3 * (a) * (b)² = + 3ab²
• Plus the cube of the second term: (b)³ = b³

to³ + 3a²b + 3ab² + b³

This same rule applies to all binomials that are cubed.

Examples of binomial cubed

Example 1.- (x + y)³

• First term cube: (x)³ = x³
• Plus the triple product of the square of the first by the second: + 3 * (x)²* (and) = +3x²Y
• Plus the triple product of the first by the square of the second: + 3 * (x) * (y)² = + 3xy²
• Plus the cube of the second term: (y)³ = + and³

x³ + 3x²y + 3xy² + and³

Example 2.- (x - y)³

• First term cube: (x)³ = x³
• Plus the triple product of the square of the first by the second: + 3 * (x)²* (- and) = -3x²Y
• Plus the triple product of the first by the square of the second: + 3 * (x) * (- y)² = + 3xy²
• Plus the cube of the second term: (-y)³ = -Y³

x³ - 3x²y + 3xy² - Y³

Example 3.- (x + ab)³

• First term cube: (x)³ = x³
• Plus the triple product of the square of the first by the second: + 3 * (x)²* (ab) = +3abx²
• Plus the triple product of the first by the square of the second: + 3 * (x) * (ab)² = + 3a²b²x
• Plus the cube of the second term: (ab)³ = + a³b³

x³ + 3abx² + 3a²b²x + a³b³

Example 4.- (and - cd)³

• First term cube: (y)³ = Y³
• Plus the triple product of the square of the first by the second: + 3 * (y)²* (- cd) = -3cdy²
• Plus the triple product of the first by the square of the second: + 3 * (y) * (- cd)² = + 3c²d²Y
• Plus the cube of the second term: (-cd)³ = -c³d³

Y³ - 3cdy² + 3c²d²y - c³d³

Example 5.- (2x + z)³

• First term cube: (2x)³ = 8x³
• Plus the triple product of the square of the first by the second: + 3 * (2x)²* (z) = +12x²z
• Plus the triple product of the first by the square of the second: + 3 * (2x) * (z)² = + 6xz²
• Plus the cube of the second term: (z)³ = + z³

8x³ + 12x²z + 6xz² + z³

Example 6.- (x - 2y)³

• First term cube: (x)³ = x³
• Plus the triple product of the square of the first by the second: + 3 * (x)²* (- 2y) = -6x²Y
• Plus the triple product of the first by the square of the second: + 3 * (x) * (- 2y)² = + 12xy²
• Plus the cube of the second term: (-2y)³ = -8y³

x³ - 6x²and + 12xy² - 8y³

Example 7.- (to²b + x)³

• First term cube: (a²b)³ = to⁶b³
• Plus the triple product of the square of the first by the second: + 3 * (a²b)²* (x) = +3rd⁴b²x
• Plus the triple product of the first by the square of the second: + 3 * (a²b) * (x)² = + 3a²bx²
• Plus the cube of the second term: (x)³ = x³

to⁶b³ + 3a⁴b²x + 3a²bx² + x³

Example 8.- (ab² + and)³

• Cube of the first term: (ab²)³ = to³b⁶
• Plus the triple product of the square of the first by the second: + 3 * (ab²)²* (and) = +3rd²b⁴Y
• Plus the triple product of the first by the square of the second: + 3 * (ab²)*(Y)² = + 3ab²Y²
• Plus the cube of the second term: (y)³ = Y³

to³b⁶ + 3a²b⁴and + 3ab²Y²+ and³

Example 9.- (x³ + and²)³

• Cube of the first term: (x³)³ = x⁹
• Plus the triple product of the square of the first by the second: + 3 * (x³)²*(Y²) = +3x⁶Y²
• Plus the triple product of the first by the square of the second: + 3 * (x³)*(Y²)² = + 3x³Y⁴
• Plus the cube of the second term: (and²)³ = Y⁶

x⁹ + 3x⁶Y² + 3x³Y⁴+ and⁶

Example 10.- (xy²z - a)³

• Cube of the first term: (xy²z)³ = x³Y⁶z³
• Plus the triple product of the square of the first by the second: + 3 * (xy²z)²(-a) = -3ax²Y⁴z²
• Plus the triple product of the first by the square of the second: + 3 * (xy²z) (- a)² = + 3a²xy²z
• Plus the cube of the second term: (-a)³ = -to³

x³Y⁶z³ -3ax²Y⁴z² + 3a²xy²z - a³