(x%2B1)(x%2B2)%2B2.jpeg "(x^2+3x+5)(x+1)(x+2)+2")
Expanding and Simplifying the Expression (x^2 + 3x + 5)(x + 1)(x + 2) + 2
This article will guide you through the process of expanding and simplifying the algebraic expression (x^2 + 3x + 5)(x + 1)(x + 2) + 2.
Step 1: Expanding the First Two Factors
We begin by expanding the first two factors: (x^2 + 3x + 5)(x + 1). We can do this using the distributive property (also known as FOIL method).
- Multiply x^2 by (x + 1): x^2 * (x + 1) = x^3 + x^2
- Multiply 3x by (x + 1): 3x * (x + 1) = 3x^2 + 3x
- Multiply 5 by (x + 1): 5 * (x + 1) = 5x + 5
Combining these results, we get:
(x^2 + 3x + 5)(x + 1) = x^3 + x^2 + 3x^2 + 3x + 5x + 5 = x^3 + 4x^2 + 8x + 5
Step 2: Expanding the Entire Expression
Now we have: (x^3 + 4x^2 + 8x + 5)(x + 2) + 2. We repeat the distributive property:
- Multiply x^3 by (x + 2): x^3 * (x + 2) = x^4 + 2x^3
- Multiply 4x^2 by (x + 2): 4x^2 * (x + 2) = 4x^3 + 8x^2
- Multiply 8x by (x + 2): 8x * (x + 2) = 8x^2 + 16x
- Multiply 5 by (x + 2): 5 * (x + 2) = 10x + 10
Combining these and adding the constant term 2, we get:
(x^3 + 4x^2 + 8x + 5)(x + 2) + 2 = x^4 + 2x^3 + 4x^3 + 8x^2 + 8x^2 + 16x + 10x + 10 + 2 = x^4 + 6x^3 + 16x^2 + 26x + 12
Conclusion
Therefore, the simplified form of the expression (x^2 + 3x + 5)(x + 1)(x + 2) + 2 is x^4 + 6x^3 + 16x^2 + 26x + 12.