Exam 3: Polynomial and Rational Functions

Translate the given formula to an English phrase using the word ʺvariesʺ. - $\mathrm { f } -$ stop $= \frac { \mathrm { f } } { \mathrm { D } }$ , where $\mathrm { f } -$ stop is camera setting with a lens with focal length $\mathrm { f }$ and diaphragm opening D

Express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. - $f ( x ) = 2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 3 x ; k = - 3$

Graph the polynomial function. Factor first if the expression is not in factored form. - $f(x)=x^{3}+3 x^{2}-x-3$

Solve the problem. -The number of mosquitoes $\mathrm { M } ( \mathrm { x } )$ , in millions, in a certain area depends on the June rainfall $\mathrm { x }$ , in inches: $\mathrm { M } ( \mathrm { x } ) = 13 \mathrm { x } - \mathrm { x } ^ { 2 }$ . What rainfall produces the maximum number of mosquitoes?

Solve the problem. -The length of a rectangle is 4 inches more than its width. If 2 inches are taken from the length and added to the width, the figure becomes a square with an area of 144 square inches. What are the dimensions of the original figure?

Translate the given formula to an English phrase using the word ʺvariesʺ. - $\mathrm { r } = \frac { \mathrm { d } } { \mathrm { t } } \text {, where } \mathrm { r } \text { is the rate by which distance } \mathrm { d } \text { is covered in time }$

Find the correct end behavior diagram for the given polynomial function. - $P ( x ) = 3 x ^ { 7 } + 6 x ^ { 2 } - 9$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $- \sqrt { 5 } , \sqrt { 5 }$ , and 1 (multiplicity 2 )

Use synthetic division to decide whether the given number k is a zero of the given polynomial function. - $- 2 ; f ( x ) = - 2 x ^ { 3 } + x ^ { 2 } - 4 x + 3$

Find the zeros of the polynomial function and state the multiplicity of each. - $f ( x ) = - 5 x ^ { 2 } ( x - 8 ) ( x + 4 ) ^ { 3 }$

Give all possible rational zeros for the following polynomial. - $P ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3$

Factor f(x) into linear factors given that k is a zero of f(x). - $\mathrm { f } ( \mathrm { x } ) = \mathrm { x } ^ { 3 } - 75 \mathrm { x } - 250 ; \mathrm { k } = - 5$ (multiplicity 2 $)$

Solve the problem. - $f(x)=-x^{4}+4$

Use a graphing calculator to find the coordinates of the turning points of the graph of the polynomial function in the indicated domain interval. Give answers to the nearest hundredth. - $f ( x ) = x ^ { 3 } - 2 x + 3 ; [ 0,1 ]$

Determine whether the statement is true or false. -If f(x) is a polynomial having only real coefficients and 2 - 4i is a zero of f(x), then 2 + 4i is also a zero.

Use the remainder theorem and synthetic division to find f(k). - $\mathrm { k } = 5 ; \mathrm { f } ( \mathrm { x } ) = \mathrm { x } ^ { 3 } - 4 \mathrm { x } ^ { 2 } + 5 \mathrm { x } - 1$

Answer the question -How can the graph of $f ( x ) = \frac { 1 } { x - 2 } + 10$ be obtained from the graph of $y = \frac { 1 } { x }$ ?

Solve the problem. Round to the nearest tenth unless indicated otherwise. -Wind resistance or atmospheric drag tends to slow down moving objects. Atmospheric drag varies jointly as an object's surface area A and velocity v. If a car traveling at a speed of $30 \mathrm { mph }$ with a surface area of $31 \mathrm { ft } ^ { 2 }$ experiences a drag of $130.2 \mathrm {~N}$ (Newtons), how fast must a car with $50 \mathrm { ft } ^ { 2 }$ of surface area travel in order to experience a drag force of $525 \mathrm {~N}$ ?

Determine whether the statement is true or false. -If f(x) is a polynomial and f(-5) = 0, then x + 5 is a factor of f(x).

Solve the problem. -If $y$ varies directly as $x ^ { 2 }$ and inversely as $m$ , and $y = 6$ when $x = 8$ and $m = 4$ , find $y$ when $x = 9$ and $m = 7 .$