(c%2Bd)(e%2Bf).jpeg "(a+b)(c+d)(e+f)")
Expanding (a+b)(c+d)(e+f)
The expression (a+b)(c+d)(e+f) represents the product of three binomials. Expanding this expression means multiplying out all the terms to get a single polynomial. This can be done systematically using the distributive property of multiplication.
Step-by-Step Expansion
-
Expand the first two binomials: (a+b)(c+d) = ac + ad + bc + bd
-
Multiply the result by the third binomial: (ac + ad + bc + bd)(e+f) = ace + acf + ade + adf + bce + bcf + bde + bdf
Final Result
The fully expanded form of (a+b)(c+d)(e+f) is:
ace + acf + ade + adf + bce + bcf + bde + bdf
Key Observations
- Number of terms: Notice that the expanded form has 8 terms, which is consistent with the fact that we are multiplying 3 binomials.
- Variables: Each term in the expanded form contains all the variables (a, b, c, d, e, f) from the original expression.
- Coefficients: The coefficients of each term are all 1.
Applications
Expanding expressions like (a+b)(c+d)(e+f) is a fundamental skill in algebra. It is used in various contexts, including:
- Solving equations: Expanding can help simplify equations and make them easier to solve.
- Factoring: Understanding the expansion process can be helpful when factoring expressions into their binomial factors.
- Calculus: Expanding expressions is often used in calculus to find derivatives and integrals.
Example
Let's consider a numerical example:
(2+3)(4+5)(6+7) = (5)(9)(13) = 585
This illustrates how expanding the expression can be used to calculate the product of multiple binomials.
Conclusion
Expanding (a+b)(c+d)(e+f) provides a concise polynomial representation of the product of three binomials. This process is a crucial step in many algebraic calculations and demonstrates the power of the distributive property.