Expand (1-x)^4. Hence find S if S = (1-x^3)^4 - 4(1-x^3)^3 + 6 (1-x^3)^2 - 4(1-x^3) + 1.
S = (1-(1-x^3))^4 = (x^3)^4 = x^12
To expand (1-x)^4, we can use the binomial theorem. The binomial theorem states that for any real numbers a and b, and any positive integer n:
(a + b)^n = C(n,0)a^n b^0 + C(n,1)a^(n-1) b^1 + C(n,2)a^(n-2) b^2 + ... + C(n,n-1)a^1 b^(n-1) + C(n,n)a^0 b^n
where C(n,k) represents the binomial coefficient, which is given by:
C(n,k) = n! / (k! (n-k)!)
Now let's apply the binomial theorem to (1-x)^4:
(1-x)^4 = C(4,0) * 1^4 * (-x)^0 + C(4,1) *1^3 * (-x)^1 + C(4,2) * 1^2 * (-x)^2 + C(4,3) * 1^1 * (-x)^3 + C(4,4) * 1^0 * (-x)^4
Simplifying each term:
(1-x)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4
We can substitute this expansion into the expression for S:
S = (1-x^3)^4 - 4(1-x^3)^3 + 6(1-x^3)^2 - 4(1-x^3) + 1
Replacing (1-x^3) with a variable, let's say y:
S = y^4 - 4y^3 + 6y^2 - 4y + 1
Now we can substitute the expansion of (1-x)^4 into our expression for S:
S = (1 - x^3)^4 - 4(1 - x^3)^3 + 6(1 - x^3)^2 - 4(1 - x^3) + 1
S = (1 - x^3 + 6x^6 - 4x^9 + x^12) - 4(1 - x^3 + 3x^6 - 3x^9 + x^12) + 6(1 - x^3 + 2x^6 - x^9) - 4(1 - x^3) + 1
Expanding each term:
S = 1 - x^3 + 6x^6 - 4x^9 + x^12 - 4 + 4x^3 - 12x^6 + 12x^9 - 4x^12 + 6 - 6x^3 + 12x^6 - 6x^9 - 4 + 4x^3 - 1
Simplifying and combining like terms:
S = -4x^12 + 3x^9 + 4x^6 - 5x^3 + 4
So, the value of S is -4x^12 + 3x^9 + 4x^6 - 5x^3 + 4.
To expand (1-x)^4, we can use the binomial theorem, which states that for any real numbers a and b and any positive integer n:
(a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^(n-1) b^1 + C(n, 2)a^(n-2) b^2 + ... + C(n, n-1)a^1 b^(n-1) + C(n, n)a^0 b^n
where C(n, k) denotes the binomial coefficient, given by:
C(n, k) = n! / (k! * (n-k)!)
For (1-x)^4, we substitute a = 1 and b = -x, and n = 4:
(1 + (-x))^4 = C(4, 0)1^4 (-x)^0 + C(4, 1)1^3 (-x)^1 + C(4, 2)1^2 (-x)^2 + C(4, 3)1^1 (-x)^3 + C(4, 4)1^0 (-x)^4
Simplifying, we have:
(1-x)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4
Now, let's substitute this result into the expression for S:
S = (1-x^3)^4 - 4(1-x^3)^3 + 6(1-x^3)^2 - 4(1-x^3) + 1
Replacing (1-x^3) with a variable y, we can rewrite S as:
S = y^4 - 4y^3 + 6y^2 - 4y + 1
Now, using the expanded form of (1-x)^4, we substitute y = (1-x^3):
S = (1-x^3)^4 - 4(1-x^3)^3 + 6(1-x^3)^2 - 4(1-x^3) + 1
= ((1-x^3) - 4(1-x^3)^2 + 6(1-x^3)^2 - 4(1-x^3) + 1)^4
Expanding each term, we get:
S = (1 - 4(1-x^3) + 6(1-x^3)^2 - 4(1-x^3)^3 + (1-x^3))^4
= (1 - 4 + 4x^3 + 6 - 12x^3 + 6x^6 + 4 - 4x^3 + x^6)^4
Now, we simplify further:
S = (8x^6 - 20x^3 + 4)^4
So, the expression for S is (8x^6 - 20x^3 + 4)^4.