question:The polynomial x^3 Ax + 15 has three real roots. Two of these roots sum to 5. What is |A|? Relevant Theorem: Vieta's formulas, named after the French mathematician François Viète, are a set of algebraic equations that relate the coefficients of a polynomial to the sums and products of its roots. These formulas are particularly useful in solving polynomial equations and finding relationships between the roots without actually calculating the roots themselves. Consider a polynomial equation of degree n: P(x) = a_nx^n + a_(n1)x^(n1) + ... + a_1x + a_0 where a_n, a_(n1), ..., a_1, and a_0 are the coefficients of the polynomial, and x is the variable. Let r_1, r_2, ..., r_n be the roots of the polynomial, i.e., P(r_i) = 0 for i = 1, 2, ..., n. Vieta's formulas establish the following relationships between the coefficients and the roots: 1. Sum of the roots: r_1 + r_2 + ... + r_n = a_(n1) / a_n 2. Sum of the products of the roots taken two at a time: r_1r_2 + r_1r_3 + ... + r_(n1)r_n = a_(n2) / a_n 3. Sum of the products of the roots taken three at a time: r_1r_2r_3 + r_1r_2r_4 + ... + r_(n2)r_(n1)r_n = a_(n3) / a_n And so on, until the product of all the roots: 4. Product of the roots: r_1r_2...r_n = (1)^n * (a_0 / a_n) These formulas can be applied to various problems in algebra, such as finding the roots of a polynomial, solving systems of equations, and simplifying expressions involving roots.

question:What's the value of a > 0, such that the tangent line to the graph of f(x) = (x^2) (e^(x)) at x = a passes through the origin? Relevant Theorem: The Derivative Chain Rule is a fundamental rule in calculus used to find the derivative of a composite function. A composite function is a function that is formed by combining two or more functions, where the output of one function becomes the input of another function. The Chain Rule states that if you have a composite function, say h(x) = f(g(x)), then the derivative of h(x) with respect to x, denoted as h'(x) or dh/dx, can be found by taking the derivative of the outer function f with respect to the inner function g(x), and then multiplying it by the derivative of the inner function g(x) with respect to x. Mathematically, the Chain Rule can be expressed as: h'(x) = f'(g(x)) * g'(x) or dh/dx = (df/dg) * (dg/dx) The Chain Rule is particularly useful when dealing with complex functions that involve multiple layers of functions, as it allows us to break down the problem into simpler parts and find the derivative step by step.

question:For a simple random walk S_n with S_0=0 and P(S_nS_{n1}=1)=1/4, P(S_nS_{n1}=1)=3/4. Let M=max{S_n:ngeq 0}. What is the probability of the event {Mgeq 5}? Round the answer to the thousands decimal. Relevant Theorem: Random walk is a mathematical concept in probability theory that describes a path consisting of a series of random steps. It is a stochastic process, meaning it involves a sequence of random variables, where each variable represents a step in the walk. The random walk can occur in onedimensional, twodimensional, or even higherdimensional spaces. In a random walk, an object, often referred to as a walker, starts at an initial position and moves in a series of steps, with each step being determined by a random variable. The direction and distance of each step are typically drawn from a probability distribution, which can be uniform, Gaussian, or any other distribution depending on the problem being modeled. A simple example of a random walk is a onedimensional random walk on a number line, where a walker starts at position 0 and at each step, moves either one step to the left or one step to the right with equal probability. After a certain number of steps, the walker's position can be anywhere on the number line, and the probability of being at a particular position can be calculated. Random walks have applications in various fields, including physics, biology, economics, and computer science. They are used to model phenomena such as stock market fluctuations, diffusion processes, animal foraging behavior, and even the movement of molecules in a fluid. Random walks also play a crucial role in the development of algorithms for search and optimization problems.

question:While a person is walking, his arms swing through approximately a 45° angle in 0.5s.As a reasonable approximation, we can assume that the arm moves with constant speed during each swing. A typical arm is 70.0 cm long, measured from the shoulder joint. What is the acceleration (in metre per second squared) of a 1.0 g drop of blood in the fingertips at the bottom of the swing? Relevant Theorem: Newton's Laws of Motion, also known as Newton's Kinetics, are three fundamental principles that describe the relationship between the motion of an object and the forces acting upon it. These laws laid the foundation for classical mechanics and have been widely used to understand and predict the behavior of objects in motion. The three laws are as follows: 1. Newton's First Law (Law of Inertia): This law states that an object at rest will stay at rest, and an object in motion will stay in motion with a constant velocity, unless acted upon by an external force. In other words, an object will maintain its state of rest or uniform motion in a straight line unless a force is applied to change its state. 2. Newton's Second Law (Law of Acceleration): This law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, it can be expressed as F = ma, where F is the net force acting on the object, m is its mass, and a is the acceleration. This means that when a force is applied to an object, it will cause the object to accelerate in the direction of the force, and the acceleration will be greater for objects with smaller mass. 3. Newton's Third Law (Action and Reaction): This law states that for every action, there is an equal and opposite reaction. In other words, when an object exerts a force on another object, the second object exerts an equal and opposite force back on the first object. This principle helps explain various phenomena, such as the recoil of a gun when fired or the propulsion of a rocket. In summary, Newton's Laws of Motion (Kinetics) provide a fundamental framework for understanding the relationship between forces and the motion of objects, which has been essential in the development of physics and engineering.