(c%2Bd)(e%2Bf)-formula.jpeg "(a+b)(c+d)(e+f) Formula")
The (a+b)(c+d)(e+f) Formula: Expanding Trinomial Products
The expression (a+b)(c+d)(e+f) represents the product of three binomials. Expanding this formula involves multiplying each term in the first binomial by each term in the second binomial, and then multiplying each of those results by each term in the third binomial. This can be a bit tedious, but there's a systematic approach to ensure you get the correct result.
Understanding the Distributive Property
The key to expanding this expression lies in the distributive property. It states that multiplying a sum by a number is the same as multiplying each term of the sum by that number:
- a(b+c) = ab + ac
We'll apply this property repeatedly to expand our trinomial product.
Step-by-Step Expansion
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Expand the first two binomials:
- (a+b)(c+d) = ac + ad + bc + bd
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Multiply the result by the third binomial:
- (ac + ad + bc + bd)(e+f) = ace + acf + ade + adf + bce + bcf + bde + bdf
The Final Expanded Form
Therefore, the expanded form of (a+b)(c+d)(e+f) is:
ace + acf + ade + adf + bce + bcf + bde + bdf
Key Points to Remember:
- This formula applies to any variables or constants in place of a, b, c, d, e, and f.
- Each term in the expanded form consists of one variable from each of the three original binomials.
- There are eight terms in the expanded form. This is because there are two possibilities for each variable (either the first or second term from its corresponding binomial).
Practical Applications
This formula is useful in various algebraic manipulations and problem-solving scenarios. For example:
- Simplifying complex expressions: You can use this formula to expand and simplify expressions containing trinomial products.
- Solving equations: Expanding trinomial products can help isolate variables and solve equations.
- Factoring: Understanding how to expand trinomial products can help you recognize patterns and factor expressions.
By understanding the distributive property and applying it systematically, you can confidently expand any trinomial product of the form (a+b)(c+d)(e+f).