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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

sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak Reply

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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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