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Key equations

general form of a polynomial function f ( x ) = a n x n + ... + a 2 x 2 + a 1 x + a 0

Key concepts

  • A power function is a variable base raised to a number power. See [link] .
  • The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
  • The end behavior depends on whether the power is even or odd. See [link] and [link] .
  • A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See [link] .
  • The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See [link] .
  • The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See [link] and [link] .
  • A polynomial of degree n will have at most n x- intercepts and at most n 1 turning points. See [link] , [link] , [link] , [link] , and [link] .

Section exercises

Verbal

Explain the difference between the coefficient of a power function and its degree.

The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

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If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

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In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

As x decreases without bound, so does f ( x ) . As x increases without bound, so does f ( x ) .

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What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

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What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x , f ( x ) and as x , f ( x ) .

The polynomial function is of even degree and leading coefficient is negative.

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Algebraic

For the following exercises, identify the function as a power function, a polynomial function, or neither.

f ( x ) = ( x 2 ) 3

Power function

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f ( x ) = x 2 x 2 1

Neither

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

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f ( x ) = 3 x + 1

Neither

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For the following exercises, find the degree and leading coefficient for the given polynomial.

7 2 x 2

Degree = 2, Coefficient = –2

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2 x 2 3 x 5 + x 6  

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x ( 4 x 2 ) ( 2 x + 1 )

Degree =4, Coefficient = –2

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For the following exercises, determine the end behavior of the functions.

f ( x ) = x 4

As x , f ( x ) , as x , f ( x )

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f ( x ) = x 4

As x , f ( x ) , as x , f ( x )

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f ( x ) = 2 x 4 3 x 2 + x 1  

As x , f ( x ) , as x , f ( x )

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f ( x ) = 3 x 2 + x 2

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

As x , f ( x ) , as x , f ( x )

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For the following exercises, find the intercepts of the functions.

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

y -intercept is ( 0 , 12 ) , t -intercepts are ( 1 , 0 ) ; ( 2 , 0 ) ; and  ( 3 , 0 ) .

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g ( n ) = −2 ( 3 n 1 ) ( 2 n + 1 )

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f ( x ) = x 4 16

y -intercept is ( 0 , 16 ) . x -intercepts are ( 2 , 0 ) and ( 2 , 0 ) .

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f ( x ) = x ( x 2 2 x 8 )

y -intercept is ( 0 , 0 ) . x -intercepts are ( 0 , 0 ) , ( 4 , 0 ) , and ( 2 ,   0 ) .

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

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Graphical

For the following exercises, determine the least possible degree of the polynomial function shown.

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
tijani
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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
what is viscosity?
David
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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
Adjei
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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
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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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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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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