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Identifying the degree and leading coefficient of a polynomial function

Identify the degree, leading term, and leading coefficient of the following polynomial functions.

f ( x ) = 3 + 2 x 2 4 x 3 g ( t ) = 5 t 2 2 t 3 + 7 t h ( p ) = 6 p p 3 2

For the function f ( x ) , the highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, −4 x 3 . The leading coefficient is the coefficient of that term, −4.

For the function g ( t ) , the highest power of t is 5 , so the degree is 5. The leading term is the term containing that degree, 5 t 5 . The leading coefficient is the coefficient of that term, 5.

For the function h ( p ) , the highest power of p is 3 , so the degree is 3. The leading term is the term containing that degree, p 3 . The leading coefficient is the coefficient of that term, −1.

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Identify the degree, leading term, and leading coefficient of the polynomial f ( x ) = 4 x 2 x 6 + 2 x 6.

The degree is 6. The leading term is x 6 . The leading coefficient is 1.

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Identifying end behavior of polynomial functions

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term. See [link] .

Polynomial Function Leading Term Graph of Polynomial Function
f ( x ) = 5 x 4 + 2 x 3 x 4 5 x 4 Graph of f(x)=5x^4+2x^3-x-4.
f ( x ) = 2 x 6 x 5 + 3 x 4 + x 3 2 x 6 Graph of f(x)=-2x^6-x^5+3x^4+x^3.
f ( x ) = 3 x 5 4 x 4 + 2 x 2 + 1 3 x 5 Graph of f(x)=3x^5-4x^4+2x^2+1.
f ( x ) = 6 x 3 + 7 x 2 + 3 x + 1 6 x 3 Graph of f(x)=-6x^3+7x^2+3x+1.

Identifying end behavior and degree of a polynomial function

Describe the end behavior and determine a possible degree of the polynomial function in [link] .

Graph of an odd-degree polynomial.

As the input values x get very large, the output values f ( x ) increase without bound. As the input values x get very small, the output values f ( x ) decrease without bound. We can describe the end behavior symbolically by writing

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

In words, we could say that as x values approach infinity, the function values approach infinity, and as x values approach negative infinity, the function values approach negative infinity.

We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.

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Describe the end behavior, and determine a possible degree of the polynomial function in [link] .

Graph of an even-degree polynomial.

As x ,   f ( x ) ;   a s   x ,   f ( x ) . It has the shape of an even degree power function with a negative coefficient.

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Identifying end behavior and degree of a polynomial function

Given the function f ( x ) = 3 x 2 ( x 1 ) ( x + 4 ) , express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Obtain the general form by expanding the given expression for f ( x ) .

f ( x ) = −3 x 2 ( x 1 ) ( x + 4 ) = −3 x 2 ( x 2 + 3 x 4 ) = −3 x 4 9 x 3 + 12 x 2

The general form is f ( x ) = −3 x 4 9 x 3 + 12 x 2 . The leading term is −3 x 4 ; therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is

as   x ,   f ( x )   as   x ,   f ( x )
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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?
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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
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what is inorganic
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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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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
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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.
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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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progressive wave
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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, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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