<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 2 x 2 9 x 5 ) ÷ ( x 5 )

Got questions? Get instant answers now!

( 3 x 2 + 23 x + 14 ) ÷ ( x + 7 )

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

Got questions? Get instant answers now!

( 4 x 2 10 x + 6 ) ÷ ( 4 x + 2 )

Got questions? Get instant answers now!

( 6 x 2 25 x 25 ) ÷ ( 6 x + 5 )

x 5 , quotient: x 5 , remainder: 0

Got questions? Get instant answers now!

( x 2 1 ) ÷ ( x + 1 )

Got questions? Get instant answers now!

( 2 x 2 3 x + 2 ) ÷ ( x + 2 )

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

Got questions? Get instant answers now!

( x 3 126 ) ÷ ( x 5 )

Got questions? Get instant answers now!

( 3 x 2 5 x + 4 ) ÷ ( 3 x + 1 )

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

Got questions? Get instant answers now!

( x 3 3 x 2 + 5 x 6 ) ÷ ( x 2 )

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

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

( 3 x 3 2 x 2 + x 4 ) ÷ ( x + 3 )

Got questions? Get instant answers now!

( 2 x 3 6 x 2 7 x + 6 ) ÷ ( x 4 )

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

Got questions? Get instant answers now!

( 6 x 3 10 x 2 7 x 15 ) ÷ ( x + 1 )

Got questions? Get instant answers now!

( 4 x 3 12 x 2 5 x 1 ) ÷ ( 2 x + 1 )

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 3 x 3 2 x 2 + x 4 ) ÷ ( x + 3 )

3 x 2 11 x + 34 106 x + 3

Got questions? Get instant answers now!

( 6 x 3 + x 2 4 ) ÷ ( 2 x 3 )

Got questions? Get instant answers now!

( 2 x 3 + 7 x 2 13 x 3 ) ÷ ( 2 x 3 )

x 2 + 5 x + 1

Got questions? Get instant answers now!

( 3 x 3 5 x 2 + 2 x + 3 ) ÷ ( x + 2 )

Got questions? Get instant answers now!

( 4 x 3 5 x 2 + 13 ) ÷ ( x + 4 )

4 x 2 21 x + 84 323 x + 4

Got questions? Get instant answers now!

( x 3 3 x + 2 ) ÷ ( x + 2 )

Got questions? Get instant answers now!

( x 3 21 x 2 + 147 x 343 ) ÷ ( x 7 )

x 2 14 x + 49

Got questions? Get instant answers now!

( x 3 15 x 2 + 75 x 125 ) ÷ ( x 5 )

Got questions? Get instant answers now!

( 9 x 3 x + 2 ) ÷ ( 3 x 1 )

3 x 2 + x + 2 3 x 1

Got questions? Get instant answers now!

( 6 x 3 x 2 + 5 x + 2 ) ÷ ( 3 x + 1 )

Got questions? Get instant answers now!

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

x 3 3 x + 1

Got questions? Get instant answers now!

( x 4 3 x 2 + 1 ) ÷ ( x 1 )

Got questions? Get instant answers now!

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

x 3 x 2 + 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

x 3 6 x 2 + 12 x 8

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

x 3 9 x 2 + 27 x 27

Got questions? Get instant answers now!

( 4 x 4 2 x 3 4 x + 2 ) ÷ ( 2 x 1 )

Got questions? Get instant answers now!

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

2 x 3 2 x + 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Yes ( x 2 ) ( 3 x 3 5 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

x 2 , 4 x 4 15 x 2 4

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

No

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

Factor is x 2 + 2 x + 5

Graph of a polynomial that has a x-intercept at 2.
Got questions? Get instant answers now!

Factor is x 2 + x + 1

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

3 x 3 + 2 x 5 x 1

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

Got questions? Get instant answers now!

4 x 3 x 2 12 x + 4

Got questions? Get instant answers now!

x 4 22 x + 2

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

Got questions? Get instant answers now!

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 ?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

x + 1 x + i

1 + 1 i x + i

Got questions? Get instant answers now!

x 3 + 1 x i

x 2 i x 1 + 1 i x i

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

2 x 2 + 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

x + 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

3 x 2 2

Got questions? Get instant answers now!

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask