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Dividing Polynomials

For the following exercises, use long division to find the quotient and remainder.

x 3 2 x 2 + 4 x + 4 x 2

x 2 + 4 with remainder 12

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3 x 4 4 x 2 + 4 x + 8 x + 1

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For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

x 3 2 x 2 + 5 x 1 x + 3

x 2 5 x + 20 61 x + 3

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2 x 3 + 6 x 2 11 x 12 x + 4

2 x 2 2 x 3 , so factored form is ( x + 4 ) ( 2 x 2 2 x 3 )

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3 x 4 + 3 x 3 + 2 x + 2 x + 1

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Zeros of Polynomial Functions

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.

2 x 3 3 x 2 18 x 8 = 0

{ 2 ,   4 ,   1 2 }

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3 x 3 + 11 x 2 + 8 x 4 = 0

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2 x 4 17 x 3 + 46 x 2 43 x + 12 = 0

{ 1 ,   3 ,   4 ,   1 2 }

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4 x 4 + 8 x 3 + 19 x 2 + 32 x + 12 = 0

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For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

x 3 3 x 2 2 x + 4 = 0

0 or 2 positive, 1 negative

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2 x 4 x 3 + 4 x 2 5 x + 1 = 0

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Rational Functions

For the following exercises, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph of the function.

f ( x ) = x + 2 x 5

Intercepts ( –2 , 0 ) and ( 0 , 2 5 ) , Asymptotes x = 5 and y = 1.

Graph of f(x)=(x+1)/(x-5).

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f ( x ) = x 2 + 1 x 2 4

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f ( x ) = 3 x 2 27 x 2 + x 2

Intercepts (3, 0), (-3, 0), and ( 0 , 27 2 ) , Asymptotes x = 1 ,   x = 2 ,   y = 3.

Graph of f(x)=(3x^2-27)/(x^2+x-2).

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f ( x ) = x + 2 x 2 9

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For the following exercises, find the slant asymptote.

f ( x ) = x 2 1 x + 2

y =   x 2

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f ( x ) = 2 x 3 x 2 + 4 x 2 + 1

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Inverses and Radical Functions

For the following exercises, find the inverse of the function with the domain given.

f ( x ) = ( x 2 ) 2 , x 2

f 1 ( x ) = x + 2

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f ( x ) = ( x + 4 ) 2 3 , x 4

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f ( x ) = x 2 + 6 x 2 , x 3

f 1 ( x ) = x + 11 3

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f ( x ) = 4 x + 5 3

f 1 ( x ) = ( x + 3 ) 2 5 4 , x 3

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f ( x ) = x 3 2 x + 1

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Modeling Using Variation

For the following exercises, find the unknown value.

y varies directly as the square of x . If when x = 3 ,   y = 36 , find y if x = 4.

y = 64

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y varies inversely as the square root of x If when x = 25 ,   y = 2 , find y if x = 4.

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y varies jointly as the cube of x and as z . If when x = 1 and z = 2 , y = 6 , find y if x = 2 and z = 3.

y   =   72

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y varies jointly as x and the square of z and inversely as the cube of w . If when x = 3 , z = 4 , and w = 2 , y = 48 , find y if x = 4 , z = 5 , and w = 3.

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For the following exercises, solve the application problem.

The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

148.5 pounds

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The volume V of an ideal gas varies directly with the temperature T and inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.

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Chapter test

Give the degree and leading coefficient of the following polynomial function.

f ( x ) = x 3 ( 3 6 x 2 2 x 2 )

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Determine the end behavior of the polynomial function.

f ( x ) = 8 x 3 3 x 2 + 2 x 4

A s x , f ( x ) , a s x , f ( x )

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f ( x ) = 2 x 2 ( 4 3 x 5 x 2 )

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Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.

f ( x ) = x 2 + 2 x 8

f ( x ) = ( x + 1 ) 2 9 , vertex ( −1 , −9 ) , intercepts ( 2 , 0 ) ; ( −4 , 0 ) ; ( 0 , −8 )

Graph of f(x)=x^2+2x-8.

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Given information about the graph of a quadratic function, find its equation.

Vertex ( 2 , 0 ) and point on graph ( 4 , 12 ) .

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Solve the following application problem.

A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.

60,000 square feet

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Find all zeros of the following polynomial functions, noting multiplicities.

f ( x ) = ( x 3 ) 3 ( 3 x 1 ) ( x 1 ) 2

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f ( x ) = 2 x 6 6 x 5 + 18 x 4

0 with multiplicity 4, 3 with multiplicity 2

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Based on the graph, determine the zeros of the function and multiplicities.

Use long division to find the quotient.

2 x 3 + 3 x 4 x + 2

2 x 2 4 x + 11 26 x + 2

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Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.

x 4 + 3 x 2 4 x 2

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2 x 3 + 5 x 2 7 x 12 x + 3

2 x 2 x 4 . So factored form is ( x + 3 ) ( 2 x 2 x 4 )

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Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

f ( x ) = 2 x 3 + 5 x 2 6 x 9

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f ( x ) = 4 x 4 + 8 x 3 + 21 x 2 + 17 x + 4

1 2 (has multiplicity 2), 1 ± i 15 2

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f ( x ) = 4 x 4 + 16 x 3 + 13 x 2 15 x 18

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f ( x ) = x 5 + 6 x 4 + 13 x 3 + 14 x 2 + 12 x + 8

2 (has multiplicity 3), ± i

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Given the following information about a polynomial function, find the function.

It has a double zero at x = 3 and zeros at x = 1 and x = 2 . Its y -intercept is ( 0 , 12 ) .

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It has a zero of multiplicity 3 at x = 1 2 and another zero at x = 3 . It contains the point ( 1 , 8 ) .

f ( x ) = 2 ( 2 x 1 ) 3 ( x + 3 )

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Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.

8 x 3 21 x 2 + 6 = 0

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For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.

f ( x ) = x + 4 x 2 2 x 3

Intercepts ( 4 , 0 ) , ( 0 , 4 3 ) , Asymptotes x = 3 ,   x   = −1 ,   y = 0 .

Graph of f(x)=(x+4)/(x^2-2x-3).

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f ( x ) = x 2 + 2 x 3 x 2 4

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Find the slant asymptote of the rational function.

f ( x ) = x 2 + 3 x 3 x 1

y = x + 4

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Find the inverse of the function.

f ( x ) = 3 x 3 4

f 1 ( x ) = x + 4 3 3

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f ( x ) = 2 x + 3 3 x 1

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Find the unknown value.

y varies inversely as the square of x and when x = 3 , y = 2. Find y if x = 1.

y = 18

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y varies jointly with x and the cube root of z . If when x = 2 and z = 27 , y = 12 , find y if x = 5 and z = 8.

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Solve the following application problem.

The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?

4 seconds

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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answer
Magreth
progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Who can show me the full solution in this problem?
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Practice Key Terms 7

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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