answer:To calculate the relative frequency of each character in a given string, we can iterate over the characters in the string, count their occurrences, and then divide each count by the total number of characters to obtain the relative frequency. This relative frequency will be multiplied by 100 to obtain a percentage value. ```python def relative_frequency(string): frequency_dict = {} total_characters = len(string) for char in string: if char not in frequency_dict: frequency_dict[char] = 1 else: frequency_dict[char] += 1 relative_frequency_dict = { char: round(freq / total_characters * 100, 1) for char, freq in frequency_dict.items() } return relative_frequency_dict ```

answer:Finding the roots of a cubic polynomial can be a challenging but fascinating task. While there isn't a straightforward formula like the quadratic formula for cubic equations, we can employ various techniques to tackle this problem. One approach is to use Cardano's formula, which is a well-known method for solving cubic equations. However, this formula can be cumbersome and may involve complex numbers, even if the roots are real. Instead, let's take a more intuitive and general approach. First, recall that a cubic polynomial can have one, two, or three real roots. If we're lucky, we might be able to factor the polynomial by inspection or using synthetic division. However, this is not always possible, so we'll focus on more general methods. One useful technique is to look for a rational root using the Rational Root Theorem. This theorem states that if a rational number p/q is a root of f(x), then p must be a factor of the constant term c, and q must be a factor of the leading coefficient (in this case, 1). By testing these possible rational roots, we might be able to find one root, which can then be used to factor the polynomial and find the remaining roots. If the rational root approach doesn't work, we can consider numerical methods, such as Newton's method or the bisection method, to approximate the roots. These methods are particularly useful when the polynomial has real coefficients, as they can help us narrow down the search for roots to a specific interval. Another approach is to use the concept of depressed cubic polynomials. By performing a simple substitution, x = y - a/3, we can transform our original cubic polynomial into a depressed cubic of the form y^3 + py + q = 0. This form can be more amenable to solution by Cardano's formula or other methods. Lastly, it's worth noting that the nature of the roots can be determined by the discriminant, Δ = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2. The discriminant can tell us whether the polynomial has one real and two complex roots, or three real roots, but its calculation can be tedious. While there is no single, straightforward method for finding the roots of a cubic polynomial, a combination of these techniques can help us tackle this problem. The approach we choose will depend on the specific coefficients and the nature of the roots we're seeking.

answer:To find the sum S of the first n terms of the GP, we can use the formula for the sum of a geometric series: S = a * (1 - r^n) / (1 - r), where a is the first term and r is the common ratio. In this case, a = 2 and r = 3/2. Plugging in these values, we get: S = 2 * (1 - (3/2)^n) / (1 - 3/2). Simplifying the denominator, we have: S = 2 * (1 - (3/2)^n) / (-1/2), which can be rewritten as: S = 4 * (1 - (3/2)^n). Now, let's consider the nth term of the GP. The formula for the nth term of a GP is given by: an = a * r^(n-1). Again, substituting a = 2 and r = 3/2, we get: an = 2 * (3/2)^(n-1). It's interesting to note that as n increases, the value of (3/2)^n grows exponentially, and the sum S will either converge or diverge depending on the value of r. In this case, since r > 1, the sum S will diverge as n approaches infinity. However, for a finite value of n, the sum S will be a finite value. The formulas for S and an provide a straightforward way to calculate the sum and the nth term of the GP. For example, if we want to find the sum of the first 5 terms, we can plug in n = 5 into the formula for S. Similarly, to find the 5th term, we can substitute n = 5 into the formula for an.

answer:To solve this integral, we'll apply the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n is a constant. Breaking down the given integral into its components, we have: I = ∫(2x^2) dx + ∫(3x) dx + ∫(1) dx. Integrating each term separately, we get: ∫(2x^2) dx = (2/3)x^3 + C1 ∫(3x) dx = (3/2)x^2 + C2 ∫(1) dx = x + C3 where C1, C2, and C3 are constants of integration. Combining these results, we have: I = (2/3)x^3 + (3/2)x^2 + x + C, where C = C1 + C2 + C3. To find the definite integral, we evaluate I at the limits of integration, x = 0 and x = 2: I = [(2/3)(2)^3 + (3/2)(2)^2 + 2] - [(2/3)(0)^3 + (3/2)(0)^2 + 0] Simplifying the expression, we get: I = [(2/3)(8) + (3/2)(4) + 2] - 0 I = (16/3) + 6 + 2 I = (16/3) + 8 I = (16 + 24)/3 I = 40/3 Therefore, the value of I is 40/3.