(x%2B2)(x%2B3)(x%2B4).jpeg "(x+1)(x+2)(x+3)(x+4)")
Expanding the Expression (x+1)(x+2)(x+3)(x+4)
This article explores the expansion of the expression (x+1)(x+2)(x+3)(x+4). While it might seem daunting at first, we can break down the problem into manageable steps.
Step-by-Step Expansion:
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Pairwise Multiplication: Begin by multiplying the first two factors and the last two factors:
- (x+1)(x+2) = x² + 3x + 2
- (x+3)(x+4) = x² + 7x + 12
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Expanding the Remaining Expression: Now, we need to multiply the results from step 1:
- (x² + 3x + 2)(x² + 7x + 12)
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Distribution: To multiply these two trinomials, we can distribute each term of the first trinomial across the second trinomial:
- x²(x² + 7x + 12) + 3x(x² + 7x + 12) + 2(x² + 7x + 12)
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Simplifying: Now, expand and combine like terms:
- x⁴ + 7x³ + 12x² + 3x³ + 21x² + 36x + 2x² + 14x + 24
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Final Result: Finally, combine all the terms to get the simplified form:
- x⁴ + 10x³ + 35x² + 50x + 24
Understanding the Expansion:
Expanding this expression demonstrates the concept of polynomial multiplication. Each step involves distributing terms and combining like terms, resulting in a polynomial with a higher degree than the original factors. This process is essential in various mathematical applications, including solving equations, factoring, and analyzing functions.
Applications:
- Solving Equations: When the expression is set equal to zero, the resulting equation can be solved to find the roots or solutions.
- Factoring: The expansion helps understand how the original expression can be factored back into its individual factors.
- Calculus: Understanding the expanded form is crucial for calculating derivatives and integrals of polynomial functions.
In conclusion, expanding (x+1)(x+2)(x+3)(x+4) involves systematic multiplication and simplification. This process reveals the powerful concepts of polynomial multiplication, essential for various mathematical applications.