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Expanding the Expression (x+1)(x+3)(x+5)
This article will explore the process of expanding the expression (x+1)(x+3)(x+5). We will use the distributive property and demonstrate how to simplify the result.
Step 1: Expanding the First Two Factors
First, we'll focus on expanding the first two factors: (x+1)(x+3).
Using the distributive property (or FOIL method), we get:
(x+1)(x+3) = x(x+3) + 1(x+3)
Expanding further:
x(x+3) + 1(x+3) = x² + 3x + x + 3
Combining like terms:
x² + 3x + x + 3 = x² + 4x + 3
Step 2: Expanding the Result with the Third Factor
Now, we have the simplified expression for the first two factors: x² + 4x + 3. We need to multiply this by the third factor, (x+5).
Using the distributive property again:
(x² + 4x + 3)(x+5) = x²(x+5) + 4x(x+5) + 3(x+5)
Expanding each term:
x²(x+5) + 4x(x+5) + 3(x+5) = x³ + 5x² + 4x² + 20x + 3x + 15
Step 3: Combining Like Terms
Finally, we combine the like terms in the expression:
x³ + 5x² + 4x² + 20x + 3x + 15 = x³ + 9x² + 23x + 15
Conclusion
Therefore, the expanded form of the expression (x+1)(x+3)(x+5) is x³ + 9x² + 23x + 15. This process demonstrates the importance of the distributive property and how it simplifies complex expressions by systematically multiplying and combining terms.