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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

what is the period of cos?
SIYAMTHEMBA Reply
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Patrick
if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom

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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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