(1%2Bx-x%5E2%2Bx%5E3-x%5E4)-(1-x)(1%2Bx%2Bx%5E2%2Bx%5E3%2Bx%5E4).jpeg "(x+2)(1+x-x^2+x^3-x^4)-(1-x)(1+x+x^2+x^3+x^4)")
Simplifying the Expression: (x+2)(1+x-x^2+x^3-x^4)-(1-x)(1+x+x^2+x^3+x^4)
This expression may seem daunting at first, but we can simplify it using a few algebraic techniques. Here's how:
Recognizing Patterns
The expressions within the parentheses are special:
- (1+x-x^2+x^3-x^4): This is a finite geometric series with the first term being 1 and the common ratio being -x.
- (1+x+x^2+x^3+x^4): This is also a finite geometric series, with the first term being 1 and the common ratio being x.
Applying the Geometric Series Formula
The sum of a finite geometric series is given by:
S = a(1-r^n)/(1-r)
where:
- S is the sum of the series
- a is the first term
- r is the common ratio
- n is the number of terms
Let's apply this to our expression:
- For (1+x-x^2+x^3-x^4): a = 1, r = -x, n = 5. Therefore: S = 1(1-(-x)^5)/(1-(-x)) = (1+x^5)/(1+x)
- For (1+x+x^2+x^3+x^4): a = 1, r = x, n = 5. Therefore: S = 1(1-x^5)/(1-x)
Substituting and Simplifying
Now, let's substitute these simplified expressions back into the original expression:
(x+2)(1+x-x^2+x^3-x^4)-(1-x)(1+x+x^2+x^3+x^4) = (x+2)((1+x^5)/(1+x)) - (1-x)((1-x^5)/(1-x))
Notice that the denominators (1+x) and (1-x) cancel out:
= (x+2)(1+x^5) - (1-x)(1-x^5)
Now, we can expand the expressions:
= x + 2 + x^6 + 2x^5 - (1 - x - x^5 + x^6)
Finally, combining like terms:
= 3x + 3x^5 + 1
Conclusion
The simplified form of the expression (x+2)(1+x-x^2+x^3-x^4)-(1-x)(1+x+x^2+x^3+x^4) is 3x + 3x^5 + 1. By recognizing the patterns of geometric series and applying the formula, we were able to simplify the expression considerably.