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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

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit 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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