Conjugated Binomials Example

On algebra, a binomial is an expression with two terms, which have a different variable and are separated by a positive or negative sign. For example: a + 2b. When there is a multiplication of binomials, 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)

On this occasion, we will talk about conjugated binomials. This remarkable product is the multiplication of two binomials:

• In the first, the second term has a positive sign: (a + b)
• In the second, the second term has a negative sign: (a - b)

It is enough that the two signs are different. No matter the order.

Conjugate binomial rule

When two such binomials are multiplying, a rule will be followed to solve this operation:

• Square of the first: (a)² = a²
• Minus the square of the second: - (b)² = - b²

to² - b²

This very simple rule is verified below, multiplying the binomials in the traditional way, term by term:

(a + b) * (a - b)

• (a) * (a) = to²
• (a) * (- b) = -ab
• (b) * (a) = + ab
• (b) * (- b) = -b²

The results are put together and form the expression:

to² - ab + ab - b²

By having opposite signs, (-ab) and (+ ab) cancel each other out, leaving finally:

to² - b²

Examples of conjugated binomials

Example 1.- (x + y) * (x - y) =x² - Y²

• (x) * (x) = x²
• (x) * (- y) = -xy
• (y) * (x) = + xy
• (y) * (- y) = -Y²

The results are put together and form the expression:

x² - xy + xy - y²

By having opposite signs, (-xy) and (+ xy) cancel each other out, finally leaving:

x² - Y²

Example 2.- (a + c) * (a - c) =to² - c²

• (a) * (a) = to²
• (a) * (- c) = -ac
• (c) * (a) = + ac
• (c) * (- c) = -c²

The results are put together and form the expression:

to² - ac + ac - c²

By having opposite signs, (-ac) and (+ ac) cancel each other out, leaving finally:

to² - c²

Example 3.- (x² + and²) * (x² - Y²) =x⁴ - Y⁴

• (x²) * (x²) = x⁴
• (x²)*(-Y²) = -x²Y²
• (Y²) * (x²) = + x²Y²
• (Y²)*(-Y²) = -Y⁴

The results are put together and form the expression:

x⁴ - x²Y² + x²Y² - Y⁴

By having opposite signs, (-x²Y²) and (+ x²Y²) are canceled, leaving finally:

x⁴ - Y⁴

Example 4.- (4x + 8y²) * (4x - 8y²) =16x² - 64y⁴

• (4x) * (4x) = 16x²
• (4x) * (- 8y²) = -32xy²
• (8y²) * (4x) = + 32xy²
• (8y²) * (- 8y²) = -64y⁴

The results are put together and form the expression:

16x² - 32xy² + 32xy² - 64y⁴

By having opposite signs, (-xy) and (+ xy) cancel each other out, finally leaving:

16x² - 64y⁴

Example 5.- (x³ + 3a) * (x³ - 3a) =x⁶ - 9a²

• (x³) * (x³) = x⁶
• (x³) * (- 3a) = -3ax³
• (3a) * (x³) = + 3ax³
• (3rd) * (- 3rd) = -9a²

The results are put together and form the expression:

x⁶ - 3ax³ + 3ax³ - 9a²

By having opposite signs, (-xy) and (+ xy) cancel each other out, finally leaving:

x⁶ - 9a²

Example 6.- (a + 2b) * (a - 2b) =to² - 4b²

• (a) * (a) = to²
• (a) * (- 2b) = -2ab
• (2b) * (a) = + 2ab
• (2b) * (- 2b) = -4b²

The results are put together and form the expression:

to² - 2ab + 2ab - 4b²

By having opposite signs, (-2ab) and (+ 2ab) cancel each other out, finally being:

to² - 4b²

Example 7.- (2c + 3d) * (2c - 3d) =4c² - 9d²

• (2c) * (2c) = 4c²
• (2c) * (- 3d) = -6cd
• (3d) * (2c) = + 6cd
• (3d) * (- 3d) = -9d²

The results are put together and form the expression:

4c² - 6cd + + 6cd - 9d²

By having opposite signs, (-6cd) and (+ 6cd) cancel each other out, finally being:

4c² - 9d²