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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

how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
Practice Key Terms 2

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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