(x^2+x+1)(2x^2-x+3)-(2x^4+x^3+4x^2-x-2)-(3x-5)-3

3 min read Jun 17, 2024
(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.