Question

Раскрыть скобки а) (х+4)^6; б) (3z – 2y3)7.

Answer 1

А)6x+24.
б)21z-42y.
пожалуйста

Answer 2

Sure, let's expand the given expressions by applying the binomial theorem.

a) Expand (x + 4)^6:

The binomial theorem states that for any real numbers "a" and "b" and a non-negative integer "n," the expansion of (a + b)^n is given by:

(a + b)^n = Σ(C(n, k) * a^(n-k) * b^k), where Σ is the summation sign, k goes from 0 to n, and C(n, k) represents the binomial coefficient, which is given by C(n, k) = n! / (k! * (n - k)!).

In our case, a = x, b = 4, and n = 6, so the expanded form will be:

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

Now, let's calculate the binomial coefficients:

C(6, 0) = 6! / (0! * (6 - 0)!) = 1 C(6, 1) = 6! / (1! * (6 - 1)!) = 6 C(6, 2) = 6! / (2! * (6 - 2)!) = 15 C(6, 3) = 6! / (3! * (6 - 3)!) = 20 C(6, 4) = 6! / (4! * (6 - 4)!) = 15 C(6, 5) = 6! / (5! * (6 - 5)!) = 6 C(6, 6) = 6! / (6! * (6 - 6)!) = 1

Now, plug these coefficients into the expanded form:

(x + 4)^6 = 1 * x^6 * 4^0 + 6 * x^5 * 4^1 + 15 * x^4 * 4^2 + 20 * x^3 * 4^3 + 15 * x^2 * 4^4 + 6 * x^1 * 4^5 + 1 * x^0 * 4^6

Simplifying the terms:

(x + 4)^6 = x^6 + 6 * 4 * x^5 + 15 * 16 * x^4 + 20 * 64 * x^3 + 15 * 256 * x^2 + 6 * 1024 * x + 4096

So, the expanded form of (x + 4)^6 is: x^6 + 24x^5 + 240x^4 + 1280x^3 + 3840x^2 + 6144x + 4096.

b) Expand (3z - 2y^3)^7:

Similar to the previous example, we will use the binomial theorem with a = 3z, b = -2y^3, and n = 7:

(3z - 2y^3)^7 = Σ(C(7, k) * (3z)^(7-k) * (-2y^3)^k), where k goes from 0 to 7.

Now, let's calculate the binomial coefficients:

C(7, 0) = 7! / (0! * (7 - 0)!) = 1 C(7, 1) = 7! / (1! * (7 - 1)!) = 7 C(7, 2) = 7! / (2! * (7 - 2)!) = 21 C(7, 3) = 7! / (3! * (7 - 3)!) = 35 C(7, 4) = 7! / (4! * (7 - 4)!) = 35 C(7, 5) = 7! / (5! * (7 - 5)!) = 21 C(7, 6) = 7! / (6! * (7 - 6)!) = 7 C(7, 7) = 7! / (7! * (7 - 7)!) = 1

Now, plug these coefficients into the expanded form:

(3z - 2y^3)^7 = 1 * (3z)^7 * (-2y^3)^0 + 7 * (3z)^6 * (-2y^3)^1 + 21 * (3z)^5 * (-2y^3)^2 + 35 * (3z)^4 * (-2y^3)^3 + 35 * (3z)^3 * (-2y^3)^4 + 21 * (3z)^2 * (-2y^3)^5 + 7 * (3z)^1 * (-2y^3)^6 + 1 * (3z)^0 * (-2y^3)^7

Simplifying the terms:

(3z - 2y^3)^7 = (3z)^7 - 7 * 2 * (3z)^6 * y^3 + 21 * 4 * (3z)^5 * y^6 - 35 * 8 * (3z)^4 * y^9 + 35 * 16 * (3z)^3 * y^12 - 21 * 32 * (3z)^2 * y^15 + 7 * 64 * (3z) * y^18 - 128 * y^21

So, the expanded form of (3z - 2y^3)^7 is: 2187z^7 - 2268z^6y^3 + 7560z^5y^6 - 10080z^4y^9 + 5600z^3y^12 - 1680z^2y^15 + 448y^18 - 128y^21.