how can we show that there is no constant
in the expansion of [2x – (x^2/4)]^9
general term
= C(9,n) (2x)^(9-n) (x^2/4)^n
= C(9,n) 2^(9-n) x^(9-n) (1/4)^n x^2n)
= C(9,n) (2^(9-n)(1/4)^n x^(9+n)
to have a constant, the exponent of x^(9+n) must be zero
so 9+n = 0
n = -9
BUT, n must be a positive integer, so there is no constant term in the expansion.
To determine if there is a constant term in the expansion of the expression [2x - (x^2/4)]^9, we can use the Binomial Theorem. The Binomial Theorem provides a systematic way to expand binomial expressions, like the one given here.
The Binomial Theorem states that the expansion of (a + b)^n can be calculated 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) represents the binomial coefficient, which is calculated as:
C(n, k) = n! / ((n-k)! * k!)
Now, we can apply the Binomial Theorem to the given expression [2x - (x^2/4)]^9. Notice that the expression has the form (a - b)^n, where a = 2x and b = (x^2/4). Applying the formula, we get:
[2x - (x^2/4)]^9 = C(9, 0) * (2x)^9 * (-(x^2/4))^0 + C(9, 1) * (2x)^8 * (-(x^2/4))^1 + C(9, 2) * (2x)^7 * (-(x^2/4))^2 + ... + C(9, 9) * (2x)^0 * (-(x^2/4))^9
Now, let's evaluate each term in the expansion. Notice that only the terms with even powers of x will have the potential to be constants since the constant term arises when all the variables cancel out.
The term with an odd power of x, such as (2x)^8 * (-(x^2/4))^1, will always contain at least one x. Therefore, it cannot be a constant. We need to find the term that contains only a constant.
Let's look at the terms:
C(9, 0) * (2x)^9 * (-(x^2/4))^0 = (1) * (2x)^9 * (1) = (2^9) * x^9 = 512x^9
The term (2x)^9 * (1) contains only a constant, given by 2^9 = 512.
Thus, the expansion of [2x - (x^2/4)]^9 has a constant term, which is equal to 512.
To show that there is no constant term in the expansion of [2x - (x^2/4)]^9, we can use the Binomial Theorem. The Binomial Theorem allows us to expand the expression (a + b)^n, where n is a positive integer, into a sum of terms.
The Binomial Theorem states that the expansion of (a + b)^n can be written as:
(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
Here, C(n, k) represents the binomial coefficient, which is calculated as C(n, k) = n! / (k! * (n-k)!).
In the given expression [2x - (x^2/4)]^9, we can rewrite it as (a - b)^n, where a = 2x and b = (x^2/4).
Now we can apply the Binomial Theorem to expand it:
[2x - (x^2/4)]^9 = C(9, 0) * (2x)^9 * (-(x^2/4))^0 + C(9, 1) * (2x)^8 * (-(x^2/4))^1 + ... + C(9, 8) * (2x)^1 * (-(x^2/4))^8 + C(9, 9) * (-(x^2/4))^9
To find the constant term, we look for the term where the exponent of x is 0. However, in this expansion, notice that all terms contain at least one factor of x. Therefore, there is no constant term in the expansion [2x - (x^2/4)]^9.
Hence, it can be shown that there is no constant term in the expansion of [2x - (x^2/4)]^9.