Example of Common Term Binomials

In algebra, a binomial is an expression that has two terms, separated by a plus sign (+) or a minus sign (-). When a binomial is multiplied by another binomial, there may be different cases in which the result can be predicted, following a simple rule. These products are called remarkable products.

Among them we find:

• 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)

Each of the four already has its own rule and by following them it is easy to find the results. This time we will talk about the binomials with common term.

Rule of binomials with common term

The binomials with common term they are two binomials that are multiplying, and between which there is an equal term, and a different one. For example:

(x + 2) * (x + 3)

Common term: x

Uncommon terms: 2, 3

The rule that is followed to multiply two binomials with a common term is:

• Square of the common term
• Plus the algebraic sum of the uncommon by the common term
• Plus the product of the uncommon

With the example, this rule will be put into practice:

• Square of the common term: (x)² = x²
• Plus the algebraic sum of the uncommon by the common term: (2 + 3) * x = 5x
• Plus the product of the uncommon ones: (2 * 3) = 6

The result is in the form of a trinomial:

x² + 5x + 6

Examples of binomials with common term

Example 1: (x + 8) * (x + 4)

• Square of the common term: (x)² = x²
• Plus the algebraic sum of the uncommon by the common term: (8 + 4) * x = 12x
• Plus the product of the uncommon ones: (8 * 4) = 32

The result is in the form of a trinomial:

x² + 12x + 32

Example 2: (x - 2) * (x + 9)

• Square of the common term: (x)² = x²
• Plus the algebraic sum of the uncommon by the common term: (-2 + 9) * x = 7x
• Plus the product of the uncommon ones: (-2 * 9) = -18

The result is in the form of a trinomial:

x² + 7x - 18

Example 3: (y - 10) * (y - 6)

• Square of the common term: (and)² = Y²
• Plus the algebraic sum of the uncommon by the common term: (-10 - 6) * x = -16y
• Plus the product of the uncommon: (-10 * -6) = 60

The result is in the form of a trinomial:

Y² - 16y + 60

Example 4: (x² - 4) * (x² + 2)

• Square of the common term: (x²)² = x⁴
• Plus the algebraic sum of the uncommon by the common term: (-4 + 2) * x² = -2x²
• Plus the product of the uncommon ones: (-4 * 2) = -8

The result is in the form of a trinomial:

x⁴ - 2x² – 8

Example 5: (x³ - 1) * (x³ + 7)

• Square of the common term: (x³)² = x⁶
• Plus the algebraic sum of the uncommon by the common term: (-1 + 7) * x³ = 6x³
• Plus the product of the uncommon ones: (-1 * 7) = -7

The result is in the form of a trinomial:

x⁶ + 6x³ – 7

Example 6: (x + a) * (x + b)

• Square of the common term: (x)² = x²
• Plus the algebraic sum of the uncommon by the common term: (a + b) * x = (a + b) x
• Plus the product of the uncommon ones: (a * b) = ab

The result is in the form of a trinomial:

x² + (a + b) x + ab

Example 7: (x + y) * (x - z²)

• Square of the common term: (x)² = x²
• Plus the algebraic sum of the uncommon by the common term: (y - z²) * x = (and Z²) x
• Plus the uncommon product: (y * -z²) = -and Z²

The result is in the form of a trinomial:

x² + (y-z²)X and Z²