Question

(X-2)*6 разложить по формуле бинома Ньютона

Answer 1

$(x-2)^6=x^6+6x^5*2+15x^4*2^2+20^x^3*2^3+ \\ +15x^2*2^4+6x*2^5+2^6= \\ =x^6+12x^5+60x^4+160x^3+240x^2+192x+64$

Answer 2

To expand the expression (X-2)*6 using the binomial theorem, we first need to understand the binomial theorem:

The binomial theorem states that for any real numbers 'a' and 'b' and a positive integer 'n', the expansion of (a + b)^n can be written as the sum of the binomial coefficients multiplied by 'a' raised to a decreasing power and 'b' raised to an increasing power, where the sum of the powers is always equal to 'n'.

The formula for the binomial coefficients is: nCr = n! / (r! * (n - r)!), where n! is the factorial of 'n', r is the term number starting from 0, and nCr represents the binomial coefficient for term 'r'.

In our case, we have (X - 2)*6, which can be rewritten as (X - 2)^6. Now, let's apply the binomial theorem to expand (X - 2)^6:

The binomial theorem for (X - 2)^6 is:

(X - 2)^6 = 6C0 * X^6 * (-2)^0 + 6C1 * X^5 * (-2)^1 + 6C2 * X^4 * (-2)^2 + 6C3 * X^3 * (-2)^3 + 6C4 * X^2 * (-2)^4 + 6C5 * X^1 * (-2)^5 + 6C6 * X^0 * (-2)^6

Now, let's calculate each term:

1. 6C0 = 6! / (0! * (6 - 0)!) = 1 (X^6) * (-2)^0 = X^6 * 1 = X^6

2. 6C1 = 6! / (1! * (6 - 1)!) = 6 (X^5) * (-2)^1 = X^5 * (-2) = -2X^5

3. 6C2 = 6! / (2! * (6 - 2)!) = 15 (X^4) * (-2)^2 = X^4 * 4 = 4X^4

4. 6C3 = 6! / (3! * (6 - 3)!) = 20 (X^3) * (-2)^3 = X^3 * (-8) = -8X^3

5. 6C4 = 6! / (4! * (6 - 4)!) = 15 (X^2) * (-2)^4 = X^2 * 16 = 16X^2

6. 6C5 = 6! / (5! * (6 - 5)!) = 6 (X^1) * (-2)^5 = X * (-32) = -32X

7. 6C6 = 6! / (6! * (6 - 6)!) = 1 (X^0) * (-2)^6 = 1 * 64 = 64

Now, let's put it all together:

(X - 2)^6 = X^6 - 26 * X^5 + 15 * X^4 - 20 * X^3 + 15 * X^2 - 632 * X + 64

Simplified:

(X - 2)^6 = X^6 - 12 * X^5 + 60 * X^4 - 120 * X^3 + 120 * X^2 - 192 * X + 64

So, the expanded form of (X - 2)*6 using the binomial theorem is X^6 - 12 * X^5 + 60 * X^4 - 120 * X^3 + 120 * X^2 - 192 * X + 64.