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Instituto Tecnológico y de Estudios Superiores de MonterreyVicerrectoría de Enseñanza Media

Mathematics IIIPolynomial Lab

Name _______________________________________ Id: ___________________ Tell if the following statement is true (T) or false (F).

1) The graph of the polynomial 723)(34

++−+−= x x x x P  shows an absolute maximum

2) The independent term of the polynomial )3()2()( 2+−= x x x P 

 is 12

3) The range for every polynomial function is always all real numbers

4) If ( 2+− i x ) is a factor of a polynomial then ( 2−− i x ) is a conjugate factor of the polynomial

Based on the following graph,

5) ( )The range of the graph is the set of real numbers

6) ( ) (x – 3) is a factor of odd multiplicity

7) ( )There is an absolute maximum at x = 0

8)( )

 The zeros of the polynomial function are x = -3, x = 0 and x = 3

9) ( )The degree of the polynomial is at least degree equals 4

Based on the following graph,

10)( )

The multiplicity of the factor (x-1) is odd

11)( )

The leading coefficient is positive

12)( )

The binomial (x-3) is a factor of the polynomial

13)( )

x= 1 is a zero of the polynomial equation

14)( )

The polynomial has a relative minimum at x = 1

15)( )

The y-intercept of the graph is y = -3

Based on the following graph,

16)( )

The multiplicity of the factor x is even

17)( )

The leading coefficient is positive

18)( )

The binomial (x-4) is a factor of the polynomial

19)( )

x= 0 is a zero of the polynomial equation

20)( )

The polynomial has a relative minimum at x = 0

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Select the correct answer from the following options.

1).________ Determine the remainder for the following divisions using the Remainder Theorem. If the divisor is a factor ofthe dividend, state that.

( 5x3 - 37x2 - 68x - 36) ÷ (x - 9)

a)

4

b)

1

c)

-8

d)

0, factor 

2).________ Determine the remainder for the following divisions using the Remainder Theorem. If the divisor is a factor ofthe dividend, state that.

(x2 + 3x - 51) ÷ (x + 9)

a)

6

b)

b) 3

c)

-9

d)

0, factor 

3).__________ Select the option that shows the possible rational zeros of the function :

a) c)

b) d)

4)._________ Select the option that shows the possible rational zeros of the function :

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c)

d)

5).________ Select the polynomial function that represents the following graph

a) )1)(4()2()( 2+−−−= x x x x x P 

b)2)1)(4)(2()( −−−−= x x x x x P 

c) )1)(4)(2()( +−−−= x x x x x P 

d) )1)(4)(2()( +−−= x x x x x P 

6) ( ) Choose the option the completely determine the zeros of the polynomial  x x x x P  20)(23−−=

a) x = 0, x = 5 and x = -4 b) x = 0, x = -5, x = 4

b) x = 0, x = -4 and x = -5 d) x = -4 and x = 5

7) ( ) Which option contains all the x-coordinates of the zeros on the graph

a) x=1 ; x=- 2 ; x= - 3

b) x=- 1 ; x= 2 ; x= 3

c) x= 0 ; x= 1 ; x=-2 ; x= -3

d) x= 0 ; x= -1 ; x= 2 ; x= 3

8) ( ) Which option contains all the x-coordinates of the zeros on the graph

a) x=0 ; x=-2 ; x= -3

b) x=0 ; x= 2 ; x= 3

c) x= 0 ; x= 2 ; x= 3

2

; x=3

d) x= -2 ; x= -3 ; x=3

2

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Solve the following exercises on a second page showing all necessary procedure, frame the final answer .

I. Sketch the following polynomial using a graph utility such as g

Graphmatica, Geogebra, etc, and fill in the provided table

1) 2)

3)

4)

 

Graph Minimums or  maximums?

(coordinate)

Relative or absolute?

y-intercept x-intercept Domain( interval

notation)

Range(interval notation)

1)

2)

3)

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II. Find all real zeros of the polynomial and determine the multiplicity of each factor for each of the following

functions

5)  x x x x  f   1252)( 23−+=

6)35

4)( x x x  f   +=

7)  x x x x  f   103)(23+−−=

8) 99)1()( 2−−+= x x x x  f  

III. Find the polynomial function with real coefficients that has the given zeros.a. 2, –2i

9) 2, -2i

10)

11)

12)

IV. Given the following factors of a polynomial find the polynomial in standard form

13) x – 2 , x + 3 , x +2

14) x – 2 , x–3i, x + 3

V. In the following exercises, a) verify that the given binomial is a factor of the polynomial and b) write the

complete factorization for the polynomial

15)12872)(

234++−−= x x x x x P 

binomial ( x + 2)

16) 12161510)( 23−++−= x x x x P  binomial (x –2)

Solve.

17) Given that 124)(234

++−+−= x xkx x x P  find the value of k so that  x - 2  is a factor of P(x).

18) Given that k  x xk  x x x P  12)1(2)(223

++++−= find the value of k so that x +1 is a factor of P(X)

VII. Sketch the graph of the polynomial function showing the x-intercepts, y intercept and state the

domain.

19) P(x)= (x – 2)2 (x2 + 2x +4)

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20) P(x)= x2 (x -1)(x - 2)