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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 are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
7%2=1, 679%2=1, 866245%2=1
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply
By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
jaks
Ryan

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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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