(2x%5E2-x%2B3)-(2x%5E4%2Bx%5E3%2B4x%5E2-x-2)-(3x-5)-3.jpeg "(x^2+x+1)(2x^2-x+3)-(2x^4+x^3+4x^2-x-2)-(3x-5)-3")
Simplifying the Expression: (x^2+x+1)(2x^2-x+3)-(2x^4+x^3+4x^2-x-2)-(3x-5)-3
This article will guide you through the process of simplifying the given algebraic expression:
(x^2+x+1)(2x^2-x+3)-(2x^4+x^3+4x^2-x-2)-(3x-5)-3
Step 1: Expanding the Product
Begin by expanding the product of the first two factors:
(x^2+x+1)(2x^2-x+3)
We can use the distributive property (or FOIL method) to multiply each term in the first factor by each term in the second factor:
- x^2(2x^2-x+3) = 2x^4 - x^3 + 3x^2
- x(2x^2-x+3) = 2x^3 - x^2 + 3x
- 1(2x^2-x+3) = 2x^2 - x + 3
Adding these results gives us:
2x^4 - x^3 + 3x^2 + 2x^3 - x^2 + 3x + 2x^2 - x + 3 = 2x^4 + x^3 + 4x^2 + 2x + 3
Step 2: Combining Terms
Now, let's combine the expanded product with the remaining terms of the expression:
(2x^4 + x^3 + 4x^2 + 2x + 3) - (2x^4 + x^3 + 4x^2 - x - 2) - (3x - 5) - 3
Notice that the 2x^4, x^3, and 4x^2 terms cancel out when we distribute the negative signs:
(2x^4 - 2x^4) + (x^3 - x^3) + (4x^2 - 4x^2) + (2x + x - 3x) + (3 + 2 + 5 - 3)
Step 3: Final Simplification
This leaves us with the simplified expression:
0 + 0 + 0 + 0 + 7 = 7
Therefore, the simplified form of the expression is 7.