answer:In "Common Sense," Thomas Paine, an Englishman who recently migrated to America, presented a compelling argument that challenged the colonial mindset. Prior to the publication of this pamphlet, most colonists had primarily focused their grievances on the British Parliament, without seriously entertaining the idea of independence. Paine boldly criticized the monarchy, specifically targeting King George III, as the root cause of the unrest. He argued that the logical, or "common sense," approach was to hold the king accountable for the rebellion and to sever ties with Britain. Asserting that Americans should prioritize their own interests, Paine advocated for independence from George III's rule. He also emphasized that declaring independence would be key to gaining the support of France and Spain in a broader conflict against monarchical rule. By doing so, Paine envisaged a "holy war" that would unite nations against the British monarchy.

answer:The given polygon has a perimeter of 2.29 and an area of 0.19. The interior angles are estimated as follows (in radians): {0.73, 1.99, 3.52, 0.67, 1.87, 2.12, 1.25, 3.07, 3.13}. Note that an angle initially reported as 5.52 has been corrected to 1.87 to ensure the sum of the angles equals n-2pi for a polygon with n vertices. The polygon is classified as 'Simple'.

answer:To calculate the invariants of systems of equations based on iterating a polynomial, we can use the following steps: 1. Define P_n(x) = (P(P(P(...P(x)))-x))/(P(x)-x), where P(x) is the original polynomial and n is the number of iterations. 2. Find the roots of P_n(x)=0. These roots are the invariants of the system. 3. Calculate the sums of powers of the roots of P_n(x)=0. These sums are polynomials in the coefficients of P_n(x). 4. Use the sums of powers to calculate the invariants of the system. For example, in the case of system 1, we have: P_6(x) = (P(P(P(x)))-x)/(P(x)-x) P_6(x) = (x^8 + 4 c x^6 + 6 c^2 x^4 + 4 c^3 x^2 + c^4 - x)/(x^2 + c - x) P_6(x) = x^6 + 3 c x^4 + 3 c^2 x^2 + c^3 The roots of P_6(x)=0 are the invariants of system 1. We can calculate the sums of powers of the roots of P_6(x)=0 using the coefficients of P_6(x). sigma_1 = Sigma x_{i j} = -3 c sigma_2 = Sigma x_{i j}^2 = 3 c^2 - 2 sigma_3 = Sigma x_{i j}^3 = -3 c^3 + 3 c We can then use the sums of powers to calculate the invariants of system 1. S2 = Sigma x_{i j} P(x_{i j}) = sigma_3 + c sigma_1 = -3 c^3 + 3 c + (-3 c)(c) = 2 c - 1 S3 = Sigma x_{i j} P(x_{i j})^2 = sigma_3^2 + c sigma_2 sigma_1 = (-3 c^3 + 3 c)^2 + c (3 c^2 - 2)(-3 c) = 3 c - 1 S3r = Sigma x_{i j}^2 P(x_{i j}) = sigma_2 sigma_3 + c sigma_1^2 = (3 c^2 - 2)(-3 c^3 + 3 c) + c (-3 c)^2 = -5 c - 1 S5 = Sigma x_{i j}^2 P(x_{i j})^3 = sigma_2 sigma_3^2 + c sigma_1^3 = (3 c^2 - 2)(-3 c^3 + 3 c)^2 + c (-3 c)^3 = 3 c^2 - 3 c - 1 This method can be used to calculate the invariants of systems of equations based on iterating any polynomial.

answer:Endorphins, released during exercise, can reduce stress and are produced by the pituitary gland. These feel-good hormones contribute to a sense of well-being and can create a "runner's high," rather than making one feel sluggish and tired.