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Section exercises

Verbal

If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

The binomial is a factor of the polynomial.

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If a polynomial of degree n is divided by a binomial of degree 1, what is the degree of the quotient?

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Algebraic

For the following exercises, use long division to divide. Specify the quotient and the remainder.

( x 2 + 5 x 1 ) ÷ ( x 1 )

x + 6 + 5 x - 1 , quotient: x + 6 , remainder: 5

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( 2 x 2 9 x 5 ) ÷ ( x 5 )

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( 3 x 2 + 23 x + 14 ) ÷ ( x + 7 )

3 x + 2 , quotient:  3 x + 2 , remainder: 0

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( 4 x 2 10 x + 6 ) ÷ ( 4 x + 2 )

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( 6 x 2 25 x 25 ) ÷ ( 6 x + 5 )

x 5 , quotient: x 5 , remainder: 0

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( x 2 1 ) ÷ ( x + 1 )

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( 2 x 2 3 x + 2 ) ÷ ( x + 2 )

2 x 7 + 16 x + 2 , quotient: 2 x 7 , remainder: 16

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( x 3 126 ) ÷ ( x 5 )

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( 3 x 2 5 x + 4 ) ÷ ( 3 x + 1 )

x 2 + 6 3 x + 1 , quotient: x 2 , remainder: 6

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( x 3 3 x 2 + 5 x 6 ) ÷ ( x 2 )

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( 2 x 3 + 3 x 2 4 x + 15 ) ÷ ( x + 3 )

2 x 2 3 x + 5 , quotient: 2 x 2 3 x + 5 , remainder: 0

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For the following exercises, use synthetic division to find the quotient.

( 3 x 3 2 x 2 + x 4 ) ÷ ( x + 3 )

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( 2 x 3 6 x 2 7 x + 6 ) ÷ ( x 4 )

2 x 2 + 2 x + 1 + 10 x 4

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( 6 x 3 10 x 2 7 x 15 ) ÷ ( x + 1 )

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( 4 x 3 12 x 2 5 x 1 ) ÷ ( 2 x + 1 )

2 x 2 7 x + 1 2 2 x + 1

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( 9 x 3 9 x 2 + 18 x + 5 ) ÷ ( 3 x 1 )

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( 3 x 3 2 x 2 + x 4 ) ÷ ( x + 3 )

3 x 2 11 x + 34 106 x + 3

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( 6 x 3 + x 2 4 ) ÷ ( 2 x 3 )

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( 2 x 3 + 7 x 2 13 x 3 ) ÷ ( 2 x 3 )

x 2 + 5 x + 1

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( 3 x 3 5 x 2 + 2 x + 3 ) ÷ ( x + 2 )

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( 4 x 3 5 x 2 + 13 ) ÷ ( x + 4 )

4 x 2 21 x + 84 323 x + 4

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( x 3 3 x + 2 ) ÷ ( x + 2 )

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( x 3 21 x 2 + 147 x 343 ) ÷ ( x 7 )

x 2 14 x + 49

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( x 3 15 x 2 + 75 x 125 ) ÷ ( x 5 )

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( 9 x 3 x + 2 ) ÷ ( 3 x 1 )

3 x 2 + x + 2 3 x 1

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( 6 x 3 x 2 + 5 x + 2 ) ÷ ( 3 x + 1 )

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( x 4 + x 3 3 x 2 2 x + 1 ) ÷ ( x + 1 )

x 3 3 x + 1

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( x 4 3 x 2 + 1 ) ÷ ( x 1 )

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( x 4 + 2 x 3 3 x 2 + 2 x + 6 ) ÷ ( x + 3 )

x 3 x 2 + 2

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( x 4 10 x 3 + 37 x 2 60 x + 36 ) ÷ ( x 2 )

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( x 4 8 x 3 + 24 x 2 32 x + 16 ) ÷ ( x 2 )

x 3 6 x 2 + 12 x 8

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( x 4 + 5 x 3 3 x 2 13 x + 10 ) ÷ ( x + 5 )

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( x 4 12 x 3 + 54 x 2 108 x + 81 ) ÷ ( x 3 )

x 3 9 x 2 + 27 x 27

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( 4 x 4 2 x 3 4 x + 2 ) ÷ ( 2 x 1 )

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( 4 x 4 + 2 x 3 4 x 2 + 2 x + 2 ) ÷ ( 2 x + 1 )

2 x 3 2 x + 2

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For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

x 2 , 4 x 3 3 x 2 8 x + 4

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x 2 , 3 x 4 6 x 3 5 x + 10

Yes ( x 2 ) ( 3 x 3 5 )

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x + 3 , 4 x 3 + 5 x 2 + 8

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x 2 , 4 x 4 15 x 2 4

Yes ( x 2 ) ( 4 x 3 + 8 x 2 + x + 2 )

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x 1 2 , 2 x 4 x 3 + 2 x 1

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x + 1 3 , 3 x 4 + x 3 3 x + 1

No

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Graphical

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is x 2 x + 3

Graph of a polynomial that has a x-intercept at -1.
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Factor is ( x 2 + 2 x + 4 )

Graph of a polynomial that has a x-intercept at 1.

( x 1 ) ( x 2 + 2 x + 4 )

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Factor is x 2 + 2 x + 5

Graph of a polynomial that has a x-intercept at 2.
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Factor is x 2 + x + 1

Graph of a polynomial that has a x-intercept at 5.

( x 5 ) ( x 2 + x + 1 )

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For the following exercises, use synthetic division to find the quotient and remainder.

4 x 3 33 x 2

Quotient: 4 x 2 + 8 x + 16 , remainder: 1

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3 x 3 + 2 x 5 x 1

Quotient: 3 x 2 + 3 x + 5 , remainder: 0

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4 x 3 x 2 12 x + 4

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x 4 22 x + 2

Quotient: x 3 2 x 2 + 4 x 8 , remainder: 6

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Technology

For the following exercises, use a calculator with CAS to answer the questions.

Consider x k 1 x 1 with k = 1 ,   2 ,   3. What do you expect the result to be if k = 4 ?

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Consider x k + 1 x + 1 for k = 1 ,   3 ,   5. What do you expect the result to be if k = 7 ?

x 6 x 5 + x 4 x 3 + x 2 x + 1

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Consider x 4 k 4 x k for k = 1 ,   2 ,   3. What do you expect the result to be if k = 4 ?

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Consider x k x + 1 with k = 1 ,   2 ,   3. What do you expect the result to be if k = 4 ?

x 3 x 2 + x 1 + 1 x + 1

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Consider x k x 1 with k = 1 ,   2 ,   3. What do you expect the result to be if k = 4 ?

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Extensions

For the following exercises, use synthetic division to determine the quotient involving a complex number.

x + 1 x i

1 + 1 + i x i

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x + 1 x + i

1 + 1 i x + i

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x 3 + 1 x i

x 2 i x 1 + 1 i x i

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Real-world applications

For the following exercises, use the given length and area of a rectangle to express the width algebraically.

Length is x + 5 , area is 2 x 2 + 9 x 5.

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Length is 2 x   +   5 , area is 4 x 3 + 10 x 2 + 6 x + 15

2 x 2 + 3

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Length is 3 x 4 , area is 6 x 4 8 x 3 + 9 x 2 9 x 4

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For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.

Volume is 12 x 3 + 20 x 2 21 x 36 , length is 2 x + 3 , width is 3 x 4.

2 x + 3

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Volume is 18 x 3 21 x 2 40 x + 48 , length is 3 x 4 , width is 3 x 4.

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Volume is 10 x 3 + 27 x 2 + 2 x 24 , length is 5 x 4 , width is 2 x + 3.

x + 2

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Volume is 10 x 3 + 30 x 2 8 x 24 , length is 2 , width is x + 3.

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For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.

Volume is π ( 25 x 3 65 x 2 29 x 3 ) , radius is 5 x + 1.

x 3

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Volume is π ( 4 x 3 + 12 x 2 15 x 50 ) , radius is 2 x + 5.

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Volume is π ( 3 x 4 + 24 x 3 + 46 x 2 16 x 32 ) , radius is x + 4.

3 x 2 2

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Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
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classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
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how many types of solid did we have
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what is an ionic bond
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What is Atoms
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what is fallopian tube
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what is bladder
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what's bulbourethral gland
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urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
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what is heart
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how is urine formed in human
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how is urine formed in human
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what is the diference between a cavity and a canal
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what is the causative agent of malaria
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malaria is caused by an insect called mosquito.
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Malaria is cause by female anopheles mosquito
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Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
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a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
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stimulates the follicle to release the mature ovum into the oviduct
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what are the functions of Endocrine and pituitary gland
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endocrine secrete hormone and regulate body process
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while pituitary gland is an example of endocrine system and it's found in the Brain
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what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
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biology is the study of life.
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Biology is the study of how living organisms live and survive in a specific environment
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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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