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Access the following online resource for additional instruction and practice with graphing polynomial functions.

Key concepts

  • Polynomial functions of degree 2 or more are smooth, continuous functions. See [link] .
  • To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See [link] , [link] , and [link] .
  • Another way to find the x - intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the x - axis. See [link] .
  • The multiplicity of a zero determines how the graph behaves at the x - intercepts. See [link] .
  • The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
  • The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
  • The end behavior of a polynomial function depends on the leading term.
  • The graph of a polynomial function changes direction at its turning points.
  • A polynomial function of degree n has at most n 1 turning points. See [link] .
  • To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most n 1 turning points. See [link] and [link] .
  • Graphing a polynomial function helps to estimate local and global extremas. See [link] .
  • The Intermediate Value Theorem tells us that if f ( a )   and   f ( b ) have opposite signs, then there exists at least one value c between a and b for which f ( c ) = 0. See [link] .

Section exercises

Verbal

What is the difference between an x - intercept and a zero of a polynomial function f ?

The x - intercept is where the graph of the function crosses the x - axis, and the zero of the function is the input value for which f ( x ) = 0.

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If a polynomial function of degree n has n distinct zeros, what do you know about the graph of the function?

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Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

If we evaluate the function at a and at b and the sign of the function value changes, then we know a zero exists between a and b .

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Explain how the factored form of the polynomial helps us in graphing it.

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If the graph of a polynomial just touches the x -axis and then changes direction, what can we conclude about the factored form of the polynomial?

There will be a factor raised to an even power.

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Algebraic

For the following exercises, find the x - or t -intercepts of the polynomial functions.

C ( t ) = 2 ( t 4 ) ( t + 1 ) ( t 6 )

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C ( t ) = 3 ( t + 2 ) ( t 3 ) ( t + 5 )

( 2 , 0 ) , ( 3 , 0 ) , ( 5 , 0 )

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C ( t ) = 4 t ( t 2 ) 2 ( t + 1 )

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

( 3 , 0 ) , ( 1 , 0 ) , ( 0 , 0 )

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C ( t ) = 2 t 4 8 t 3 + 6 t 2

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C ( t ) = 4 t 4 + 12 t 3 40 t 2

( 0 , 0 ) ,   ( 5 , 0 ) ,   ( 2 , 0 )

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f ( x ) = x 4 x 2

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f ( x ) = x 3 + x 2 20 x

( 0 , 0 ) ,   ( 5 , 0 ) ,   ( 4 , 0 )

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f ( x ) = x 3 + 6 x 2 7 x

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

( 2 , 0 ) ,   ( 2 , 0 ) ,   ( 1 , 0 )

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f ( x ) = x 3 + 2 x 2 9 x 18

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f ( x ) = 2 x 3 x 2 8 x + 4

( 2 , 0 ) , ( 2 , 0 ) , ( 1 2 , 0 )

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f ( x ) = x 6 7 x 3 8

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f ( x ) = 2 x 4 + 6 x 2 8

( 1 , 0 ) ,   ( 1 , 0 )

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

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f ( x ) = x 6 2 x 4 3 x 2

( 0 , 0 ) , ( 3 , 0 ) , ( 3 , 0 )

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f ( x ) = x 6 3 x 4 4 x 2

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f ( x ) = x 5 5 x 3 + 4 x

( 0 , 0 ) ,   ( 1 , 0 ) ( 1 , 0 ) ,   ( 2 , 0 ) ,   ( 2 , 0 )

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For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

f ( x ) = x 3 9 x , between x = −4 and x = −2.

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f ( x ) = x 3 9 x , between x = 2 and x = 4.

f ( 2 ) = 10 and f ( 4 ) = 28. Sign change confirms.

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Questions & Answers

what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
Practice Key Terms 4

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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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