write down the expansion of (1+x-x^2)^6 in Ascending other please show full working!
I'll let you read up on trinomial expansions, and just refer you here:
http://www.wolframalpha.com/input/?i=%281%2Bx-x^2%29^6
scroll down a bit to see the full expanded form.
This web site is a good place to play around and explore various math topics, play around with functions, etc.
wow thanks sir
To expand the expression (1+x-x^2)^6 using the Binomial Theorem, we need to examine the pattern in the expansion coefficients and powers of each term.
The Binomial Theorem states that for any positive integer n:
(x + y)^n = C(n,0)x^n y^0 + C(n,1)x^(n-1) y^1 + C(n,2)x^(n-2) y^2 + ... + C(n,r)x^(n-r) y^r + ... + C(n,n)x^0 y^n
where C(n,r) represents the binomial coefficient, which is the number of ways to choose r items from a set of n items, given by the formula:
C(n,r) = n! / (r!(n-r)!)
Let's apply this to the given expression:
(1+x-x^2)^6
We can rewrite the expression as:
(1 - (x - x^2))^6
Now, using the Binomial Theorem, we can find the expansion coefficients and the powers of x:
The expansion will have terms of the form (ax^m)(b(-x^2)^n), where a and b are the expansion coefficients, m is the power of x, and n is the power of (-x^2).
Expanding each term using the binomial coefficients, we get:
C(6,0)(1)^(6-0)(-1)^0(x^0)(x^2)^6
+ C(6,1)(1)^(6-1)(-1)^1(x^1)(x^2)^5
+ C(6,2)(1)^(6-2)(-1)^2(x^2)(x^2)^4
+ C(6,3)(1)^(6-3)(-1)^3(x^3)(x^2)^3
+ C(6,4)(1)^(6-4)(-1)^4(x^4)(x^2)^2
+ C(6,5)(1)^(6-5)(-1)^5(x^5)(x^2)^1
+ C(6,6)(1)^(6-6)(-1)^6(x^6)(x^2)^0
Now, let's calculate each term, using the formula for binomial coefficients C(n,r):
= 1(x^0)(x^2)^6 - 6(1)(-1)(x^1)(x^2)^5 + 15(1^2)(-1^2)(x^2)(x^2)^4 - 20(1^3)(-1^3)(x^3)(x^2)^3 + 15(1^4)(-1^4)(x^4)(x^2)^2 - 6(1^5)(-1^5)(x^5)(x^2)^1 + 1(1^6)(-1^6)(x^6)(x^2)^0
Simplifying each term, we have:
= x^12 - 6x^7 + 15x^4 - 20x^5 + 15x^4 - 6x^7 + x^12
Finally, combining like terms, we get:
= 2x^12 - 12x^7 + 30x^4 - 40x^5
Therefore, the expansion of (1+x-x^2)^6 in ascending order is:
2x^12 - 12x^7 + 30x^4 - 40x^5