Exam 4: Polynomial and Rational Functions

A thin flat plate is situated in an xy-plane such that the density d ( in lb/ft ² ) at the point is inversely proportional to the square of the distance from the origin. What is the effect on the density at P if the x- and y-coordinates are each multiplied by

A thin flat plate is situated in an xy-plane such that the density d ( in lb/ft ² ) at the point is inversely proportional to the square of the distance from the origin. What is the effect on the density at P if the x- and y-coordinates are each multiplied by

Find all solutions of the equation.

Find a factored form with integer coefficients of the polynomial f shown in the figure.

A polynomial f ( x ) with real coefficients and leading coefficient 1 has the given zero and degree. Express f ( x ) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.

The pressure P acting at a point in a liquid is directly proportional to the distance d from the surface of the liquid to the point. Express P as a function of d by means of a formula that involves a constant of proportionality k. In a certain oil tank, the pressure at a depth of 7 feet is 413. Find the value of k.

Express the statement as a formula that involves the variables u, v and a constant of proportionality k, and determine the value of k from the condition : u is directly proportional to v and if v = 40, then u = 3

Find all solutions of the equation.

Use synthetic division to decide whether is a zero of . ;

Express the statement as a formula that involves the variables q, x, y and a constant of proportionality k, and then determine the value of k from the condition : q is inversely proportional to the sum of x and y, if x = 0.3 and y = 3.7, then q = 0.4

For the following function, find a number such that the graph of contains the point

Applying the first theorem on bounds for real zeros of polynomials, determine the smallest and largest integers that are upper and lower bounds, respectively, for the real solutions of the equation. With the aid of a graphing utility, discuss the validity of the bounds.

Does there exist a polynomial of degree 3 with real coefficients that has zeros 9 , - 9, and i ?

Find a polynomial of degree that has the indicated zeros and satisfies the given condition.

A canvas camping tent is to be constructed in the shape of a pyramid with a square base. An 8-foot pole will form the center support, as illustrated in the figure. Find the length x of a side of the base so that the total amount of canvas needed for the sides and bottom is 384 ft ² .

Find all values of such that

Use synthetic division to find ;

Show that the number is a zero of of the given multiplicity, and express as a product of linear factors.

Find all values of such that .

A polynomial f ( x ) with real coefficients and leading coefficient 1 has the given zero and degree. Express f ( x ) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.