(a+b+c)(d+e+f)

3 min read Jun 16, 2024

Expanding the Expression (a + b + c)(d + e + f)

The expression (a + b + c)(d + e + f) represents the product of two trinomials. To expand this expression, we can use the distributive property.

The distributive property states that for any real numbers a, b, and c:

a(b + c) = ab + ac

We can apply the distributive property twice to expand (a + b + c)(d + e + f):

Distribute the first trinomial (a + b + c) over the second trinomial (d + e + f): (a + b + c)(d + e + f) = a(d + e + f) + b(d + e + f) + c(d + e + f)
Distribute each term of the first trinomial over the second trinomial: ad + ae + af + bd + be + bf + cd + ce + cf

Therefore, the expanded form of (a + b + c)(d + e + f) is:

ad + ae + af + bd + be + bf + cd + ce + cf

The expanded form contains nine terms, each representing the product of one term from the first trinomial and one term from the second trinomial.
This expansion demonstrates the power of the distributive property in simplifying and manipulating algebraic expressions.

Let's consider an example:

If a = 2, b = 3, c = 1, d = 4, e = 5, and f = 6, then:

(a + b + c)(d + e + f) = (2 + 3 + 1)(4 + 5 + 6) = 6 * 15 = 90

We can verify this by substituting the values into the expanded form:

24 + 25 + 26 + 34 + 35 + 36 + 14 + 15 + 1*6 = 8 + 10 + 12 + 12 + 15 + 18 + 4 + 5 + 6 = 90