Answer:
Explanation:
A) The Binomial Theorem states that for any positive integer n and any real numbers a and b, the expression (a + b)^n can be expanded as a sum of the form:
(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)a^0b^n
where C(n,k) = n!/(k!(n-k)!) is the binomial coefficient.
Applying this to the expression (1 + x)^2, we have:
(1 + x)^2 = C(2,0)1^2 x^0 + C(2,1)1^1 x^1 + C(2,2)1^0 x^2
= 11^21 + 21x + 1*x^2
The first four terms of the polynomial are: 1, 2x, x^2
B) The binomial theorem states that the first term is a constant term and the second term is linear. When x is less than 1, the linear term 2x is much smaller than the constant term 1, so the linear term can be ignored as a good approximation. Therefore, 1 + nx will be a good approximation of the expression (1+x)^2 when x is less than 1.