(4n%2B1)%2B(n%2B2)(4n%2B1).jpeg "(3n+1)(4n+1)+(n+2)(4n+1)")
Expanding and Simplifying the Expression: (3n+1)(4n+1)+(n+2)(4n+1)
This article explores the process of expanding and simplifying the algebraic expression: (3n+1)(4n+1)+(n+2)(4n+1).
Expanding the Expression
To begin, we can utilize the distributive property (also known as the FOIL method) to expand each of the products within the expression:
Step 1: Expand (3n+1)(4n+1)
- 3n * 4n = 12n²
- 3n * 1 = 3n
- 1 * 4n = 4n
- 1 * 1 = 1
Combining these terms: (3n+1)(4n+1) = 12n² + 3n + 4n + 1
Step 2: Expand (n+2)(4n+1)
- n * 4n = 4n²
- n * 1 = n
- 2 * 4n = 8n
- 2 * 1 = 2
Combining these terms: (n+2)(4n+1) = 4n² + n + 8n + 2
Combining Like Terms
Now, let's combine the expanded terms from both products:
(12n² + 3n + 4n + 1) + (4n² + n + 8n + 2)
Step 3: Combine the n² terms: 12n² + 4n² = 16n² Step 4: Combine the n terms: 3n + 4n + n + 8n = 16n Step 5: Combine the constant terms: 1 + 2 = 3
Simplified Expression
Finally, we have the simplified form of the expression:
(3n+1)(4n+1)+(n+2)(4n+1) = 16n² + 16n + 3
Conclusion
By expanding and simplifying the given expression, we arrive at the polynomial expression 16n² + 16n + 3. This process highlights the importance of understanding and applying basic algebraic principles such as the distributive property and combining like terms.