Find the first four terms in the binomial expansion (x-7)^12
To find the first four terms in the binomial expansion of (x-7)^12, we can use the binomial theorem. The binomial theorem states that for any two numbers a and b, and any positive integer n, the expansion of (a + b)^n can be written as the sum of the binomial coefficients multiplied by the respective powers of a and b.
In this case, a = x, b = -7, and n = 12. We need to find the coefficients and powers of x and -7 in the expansion.
The binomial coefficients are calculated using the formula:
C(n, k) = n! / (k! * (n-k)!),
where C(n, k) denotes the binomial coefficient of n choose k, and ! represents the factorial of a number.
Let's calculate the four terms of the expansion step by step:
First term:
The coefficient of the first term is C(12, 0) = 12! / (0! * (12-0)!) = 1, as choosing 0 elements from 12 is only possible in one way (which is not choosing any elements).
The powers of x and -7 are (x)^(12-0) and (-7)^0 = 1 respectively.
Therefore, the first term is 1 * x^12 * 1 = x^12.
Second term:
The coefficient of the second term is C(12, 1) = 12! / (1! * (12-1)!) = 12, as choosing 1 element from 12 can be done in 12 ways.
The powers of x and -7 are (x)^(12-1) and (-7)^1 = -7 respectively.
Therefore, the second term is 12 * x^11 * -7 = -84x^11.
Third term:
The coefficient of the third term is C(12, 2) = 12! / (2! * (12-2)!) = 66, as choosing 2 elements from 12 can be done in 66 ways.
The powers of x and -7 are (x)^(12-2) and (-7)^2 = 49 respectively.
Therefore, the third term is 66 * x^10 * 49 = 3234x^10.
Fourth term:
The coefficient of the fourth term is C(12, 3) = 12! / (3! * (12-3)!) = 220, as choosing 3 elements from 12 can be done in 220 ways.
The powers of x and -7 are (x)^(12-3) and (-7)^3 = -343 respectively.
Therefore, the fourth term is 220 * x^9 * -343 = -75140x^9.
Hence, the first four terms in the binomial expansion of (x-7)^12 are:
x^12, -84x^11, 3234x^10, -75140x^9.
12C0 x^12 (-7)^0 + 12C1 x^11 (-7)^1 + 12C2 x^10 (-7)^2 + 12C3 x^9 (-7)^3 + ...
x^12 - 84x^11 + 3234x^10 - 75460x^9 + ...