Question

expand (1-x^2)^6 ?

Answer 1

[AI-generated answer — Explain Bot]

To expand the expression (1 - x^2)^6, we can use the binomial theorem. The binomial theorem states that for any positive integer "n", we can expand the expression (a + b)^n using the formula:

(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, defined as C(n, k) = n! / (k! * (n-k)!).

Now let's apply the binomial theorem to expand (1 - x^2)^6:

(1 - x^2)^6 = C(6, 0) * 1^6 * (-x^2)^0 + C(6, 1) * 1^5 * (-x^2)^1 + C(6, 2) * 1^4 * (-x^2)^2 + C(6, 3) * 1^3 * (-x^2)^3 + C(6, 4) * 1^2 * (-x^2)^4 + C(6, 5) * 1^1 * (-x^2)^5 + C(6, 6) * 1^0 * (-x^2)^6

Simplifying each term using the binomial coefficient and the powers, we get:

(1 - x^2)^6 = 1 - 6x^2 + 15x^4 - 20x^6 + 15x^8 - 6x^10 + x^12

Therefore, the expanded form of (1 - x^2)^6 is 1 - 6x^2 + 15x^4 - 20x^6 + 15x^8 - 6x^10 + x^12.