Polynomial Operations

You are given two polynomials P₁(x) and P₂(x) represented by their coefficients in increasing order of powers of x. That is, a polynomial P(x) = a₀ + a₁x + a₂x² + \dots + a_nxⁿ is given by the list of coefficients [a₀, a₁, ..., a_n].

Your task is to implement a Polynomial class (or equivalent in your language of choice) with the following operations:

• Addition: P(x) + Q(x)
• Subtraction: P(x) - Q(x)
• Multiplication: P(x) \times Q(x)
• Evaluation: Compute P(x) at a given value of x.
• String Representation: Convert the polynomial into a human-readable string where nonzero terms are shown in the form:
  • For the constant term: "a"
  • For x¹: "ax"
  • For higher powers: "ax^k"
  and terms are joined with " + ". If all coefficients are zero, output "0".

The input will give you the coefficients for two polynomials and an evaluation point. You are required to:

1. Construct the two polynomials P₁ and P₂ from the given coefficients.
2. Compute P₁ + P₂, P₁ - P₂, and P₁ \times P₂.
3. Evaluate both P₁ and P₂ at the given point.
4. Print the string representations of P₁, P₂, their sum, difference, product, and the evaluations.
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Note: When displaying formulas, use LaTeX for mathematical notation. For example, the polynomial should be represented as \(a_0 + a_1x + a_2x^2 + \dots + a_nx^n\).

inputFormat

The input is read from standard input (stdin) and has the following format:

n1
c11 c12 ... c1n1
n2
c21 c22 ... c2n2
x

Where:

• n1 is the number of coefficients of the first polynomial P₁.
• The next line contains n1 integers representing the coefficients of P₁ in increasing order of powers.
• n2 is the number of coefficients of the second polynomial P₂.
• The following line contains n2 integers for P₂.
• x is the evaluation point.

outputFormat

Print the following information to standard output (stdout), each on a new line:

p1: {P1_string}
p2: {P2_string}
p1 + p2: {sum_string}
p1 - p2: {diff_string}
p1 * p2: {prod_string}
p1 evaluated at x={x}: {P1_value}
p2 evaluated at x={x}: {P2_value}

Where {P1_string}, {P2_string}, etc. are the string representations and computed values of the respective polynomials.

## sample

3
1 2 3
3
0 -1 4
2

p1: 1 + 2x + 3x^2
p2: -1x + 4x^2
p1 + p2: 1 + 1x + 7x^2
p1 - p2: 1 + 3x + -1x^2
p1 * p2: -1x + 2x^2 + 5x^3 + 12x^4
p1 evaluated at x=2: 17
p2 evaluated at x=2: 14

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