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An athlete whose event is the shot put releases the shot wilh the same initial velocity but at different angles. The figure shows the parabolic paths for shots released at angles of \(35^{\circ}\) and \(65^{\circ} .\) Exercises \(57-58\) are based on the functions that model the parabolic paths. (table cannot copy) When the shot whose path is shown by the red graph is released at an angle of \(65^{\circ},\) its height, \(g(x),\) in feet, can be modeled by $$ g(x)=-0.04 x^{2}+2.1 x+6.1 $$ where \(x\) is the shot's horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through (c) and verify your answers using the red graph. a. What is the maximum height, to the nearest tenth of a foot, of the shot and how far from its point of release does this occur? b. What is the shot's maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw? c. From what height was the shot released?

How can the Division Algorithm be used to check the quotient and remainder in a long division problem?

An athlete whose event is the shot put releases the shot wilh the same initial velocity but at different angles. The figure shows the parabolic paths for shots released at angles of \(35^{\circ}\) and \(65^{\circ} .\) Exercises \(57-58\) are based on the functions that model the parabolic paths. (table cannot copy) Among all pairs of numbers whose sum is \(16,\) find a pair whose product is as large as possible. What is the maximum product?

Complex numbers are used in electronics to describe the current in an electric circuit. Ohm's law relates the current in a circuit, \(I\), in amperes, the voltage of the circuit, \(E,\) in volts, and the resistance of the circuit, \(R,\) in ohms, by the formula \(E=I R .\) Use this formula to solve. Find \(E,\) the voltage of a circuit, if \(I=(2-3 i)\) amperes and \(R=(3+5 i)\) ohms.

The mathematician Girolamo Cardano is credited with the first use (in 1545 ) of negative square roots in solving the now-famous problem, "Find two numbers whose sum is 10 and whose product is \(40 . "\) Show that the complex numbers \(5+i \sqrt{15}\) and \(5-i \sqrt{15}\) satisfy the conditions of the problem. (Cardano did not use the symbolism \(i \sqrt{15}\) or even \(\sqrt{-15} .\) He wrote R.m 15 for \(\sqrt{-15},\) meaning "radix minus 15." He regarded the numbers 5 + R.m 15 and 5 - R.m 15 as "fictitious" or "ghost numbers," and considered the problem "manifestly impossible." But in a mathematically adventurous spirit, he exclaimed, "Nevertheless, we will operate.")

Describe how to find the possible rational zeros of a polynomial function.

Describe how to use Descartes's Rule of Signs to determine the possible number of positive real zeros of a polynomial function.

What is the complex conjugate of \(2+3 i ?\) What happens when you multiply this complex number by its complex conjugate?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Every time I divide polynomials using synthetic division, I am using a highly condensed form of the long division procedure where omitting the variables and exponents does not involve the loss of any essential data.

Describe how to use Descartes's Rule of Signs to determine the possible number of negative roots of a polynomial equation.