(x%2B2)(x%2B1)%2B(x%2B2)(x%2B1)%2B(x%2B1).jpeg "(x+3)(x+2)(x+1)+(x+2)(x+1)+(x+1)")
Factoring and Simplifying the Expression (x+3)(x+2)(x+1)+(x+2)(x+1)+(x+1)
This article will guide you through the process of factoring and simplifying the expression: (x+3)(x+2)(x+1)+(x+2)(x+1)+(x+1).
Factoring by Grouping
The key to simplifying this expression lies in recognizing common factors. Notice that (x+1) appears in each term. We can factor this out:
(x+1)[(x+3)(x+2) + (x+2) + 1]
Now we can focus on simplifying the expression inside the brackets. Let's expand the first term:
(x+1)[(x^2 + 5x + 6) + (x+2) + 1]
Combine like terms:
(x+1)[x^2 + 6x + 9]
Final Simplification
The expression inside the brackets is now a perfect square trinomial: (x+3)^2
Therefore, the fully factored and simplified expression is:
(x+1)(x+3)^2
Conclusion
By strategically factoring out common terms and recognizing patterns, we've simplified the expression from a seemingly complex form to a concise and factored form: (x+1)(x+3)^2. This process demonstrates the power of algebraic manipulation in simplifying expressions and revealing their underlying structure.