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Identify Polynomials, Monomials, Binomials, and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.


81 b 5 24 b 3 + 1
5 c 3 + 11 c 2 c 8
14 15 y + 1 7
5
4 y + 17

trinomial polynomial binomial monomial binomial

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x 2 y 2
−13 c 4
x 2 + 5 x 7
x 2 y 2 2 x y + 8
19

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8 3 x
z 2 5 z 6
y 3 8 y 2 + 2 y 16
81 b 5 24 b 3 + 1
−18

binomial trinomial polynomial trinomial monomial

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11 y 2
−73
6 x 2 3 x y + 4 x 2 y + y 2
4 y + 17
5 c 3 + 11 c 2 c 8

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Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.


6 a 2 + 12 a + 14
18 x y 2 z
5 x + 2
y 3 8 y 2 + 2 y 16
−24

2 4 1 3 0

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9 y 3 10 y 2 + 2 y 6
−12 p 4
a 2 + 9 a + 18
20 x 2 y 2 10 a 2 b 2 + 30
17

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14 29 x
z 2 5 z 6
y 3 8 y 2 + 2 y 16
23 a b 2 14
−3

1 2 3 3 0

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62 y 2
15
6 x 2 3 x y + 4 x 2 y + y 2
10 9 x
m 4 + 4 m 3 + 6 m 2 + 4 m + 1

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Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

28 x ( −12 x )

40 x

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12 a + 5 b 22 a

−10a + 5 b

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2 a 2 + b 2 6 a 2

−4 a 2 + b 2

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x y 2 5 x 5 y 2

x y 2 5 x 5 y 2

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a 2 b 4 a 5 a b 2

a 2 b 4 a 5 a b 2

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Add: 4 a , −3 b , −8 a

−4 a 3 b

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Subtract 5 x 6 from 12 x 6 .

−17 x 6

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Subtract 2 p 4 from 7 p 4 .

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Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

( 5 y 2 + 12 y + 4 ) + ( 6 y 2 8 y + 7 )

11 y 2 + 4 y + 11

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( 4 y 2 + 10 y + 3 ) + ( 8 y 2 6 y + 5 )

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( x 2 + 6 x + 8 ) + ( −4 x 2 + 11 x 9 )

−3 x 2 + 17 x 1

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( y 2 + 9 y + 4 ) + ( −2 y 2 5 y 1 )

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( 8 x 2 5 x + 2 ) + ( 3 x 2 + 3 )

11 x 2 5 x + 5

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( 7 x 2 9 x + 2 ) + ( 6 x 2 4 )

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( 5 a 2 + 8 ) + ( a 2 4 a 9 )

6 a 2 4 a 1

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( p 2 6 p 18 ) + ( 2 p 2 + 11 )

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( 4 m 2 6 m 3 ) ( 2 m 2 + m 7 )

2 m 2 7 m + 4

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( 3 b 2 4 b + 1 ) ( 5 b 2 b 2 )

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( a 2 + 8 a + 5 ) ( a 2 3 a + 2 )

5 a + 3

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( b 2 7 b + 5 ) ( b 2 2 b + 9 )

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( 12 s 2 15 s ) ( s 9 )

12 s 2 14 s + 9

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( 10 r 2 20 r ) ( r 8 )

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Subtract ( 9 x 2 + 2 ) from ( 12 x 2 x + 6 ) .

3 x 2 x + 4

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Subtract ( 5 y 2 y + 12 ) from ( 10 y 2 8 y 20 ) .

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Subtract ( 7 w 2 4 w + 2 ) from ( 8 w 2 w + 6 ) .

w 2 + 3 w + 4

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Subtract ( 5 x 2 x + 12 ) from ( 9 x 2 6 x 20 ) .

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Find the sum of ( 2 p 3 8 ) and ( p 2 + 9 p + 18 ) .

2 p 3 + p 2 + 9 p + 10

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Find the sum of
( q 2 + 4 q + 13 ) and ( 7 q 3 3 ) .

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Find the sum of ( 8 a 3 8 a ) and ( a 2 + 6 a + 12 ) .

8 a 3 + a 2 2 a + 12

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Find the sum of
( b 2 + 5 b + 13 ) and ( 4 b 3 6 ) .

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Find the difference of
( w 2 + w 42 ) and
( w 2 10 w + 24 ) .

11 w 64

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Find the difference of
( z 2 3 z 18 ) and
( z 2 + 5 z 20 ) .

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Find the difference of
( c 2 + 4 c 33 ) and
( c 2 8 c + 12 ) .

12 c 45

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Find the difference of
( t 2 5 t 15 ) and
( t 2 + 4 t 17 ) .

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( 7 x 2 2 x y + 6 y 2 ) + ( 3 x 2 5 x y )

10 x 2 7 x y + 6 y 2

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( −5 x 2 4 x y 3 y 2 ) + ( 2 x 2 7 x y )

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( 7 m 2 + m n 8 n 2 ) + ( 3 m 2 + 2 m n )

10 m 2 + 3 m n 8 n 2

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( 2 r 2 3 r s 2 s 2 ) + ( 5 r 2 3 r s )

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( a 2 b 2 ) ( a 2 + 3 a b 4 b 2 )

−3 a b + 3 b 2

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( m 2 + 2 n 2 ) ( m 2 8 m n n 2 )

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( u 2 v 2 ) ( u 2 4 u v 3 v 2 )

4 u v + 2 v 2

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( j 2 k 2 ) ( j 2 8 j k 5 k 2 )

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( p 3 3 p 2 q ) + ( 2 p q 2 + 4 q 3 ) ( 3 p 2 q + p q 2 )

p 3 6 p 2 q + p q 2 + 4 q 3

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( a 3 2 a 2 b ) + ( a b 2 + b 3 ) ( 3 a 2 b + 4 a b 2 )

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( x 3 x 2 y ) ( 4 x y 2 y 3 ) + ( 3 x 2 y x y 2 )

x 3 + 2 x 2 y 5 x y 2 + y 3

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( x 3 2 x 2 y ) ( x y 2 3 y 3 ) ( x 2 y 4 x y 2 )

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Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

Evaluate 8 y 2 3 y + 2 when:

y = 5
y = −2
y = 0

187 46 2

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Evaluate 5 y 2 y 7 when:

y = −4
y = 1
y = 0

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Evaluate 4 36 x when:

x = 3
x = 0
x = −1

−104 4 40

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Evaluate 16 36 x 2 when:

x = −1
x = 0
x = 2

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A painter drops a brush from a platform 75 feet high. The polynomial −16 t 2 + 75 gives the height of the brush t seconds after it was dropped. Find the height after t = 2 seconds.

11

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A girl drops a ball off a cliff into the ocean. The polynomial −16 t 2 + 250 gives the height of a ball t seconds after it is dropped from a 250-foot tall cliff. Find the height after t = 2 seconds.

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A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial −4 p 2 + 420 p . Find the revenue received when p = 60 dollars.

$10,800

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A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial −4 p 2 + 420 p . Find the revenue received when p = 90 dollars.

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Everyday math

Fuel Efficiency The fuel efficiency (in miles per gallon) of a car going at a speed of x miles per hour is given by the polynomial 1 150 x 2 + 1 3 x . Find the fuel efficiency when x = 30 mph .

4

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Stopping Distance The number of feet it takes for a car traveling at x miles per hour to stop on dry, level concrete is given by the polynomial 0.06 x 2 + 1.1 x . Find the stopping distance when x = 40 mph .

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Rental Cost The cost to rent a rug cleaner for d days is given by the polynomial 5.50 d + 25 . Find the cost to rent the cleaner for 6 days.

$58

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Height of Projectile The height (in feet) of an object projected upward is given by the polynomial −16 t 2 + 60 t + 90 where t represents time in seconds. Find the height after t = 2.5 seconds.

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Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial 9 5 c + 32 where c represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when c = 65 ° .

149

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Writing exercises

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

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Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

Answers will vary.

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Ariana thinks the sum 6 y 2 + 5 y 4 is 11 y 6 . What is wrong with her reasoning?

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Jonathan thinks that 1 3 and 1 x are both monomials. What is wrong with his reasoning?

Answers will vary.

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “identify polynomials, monomials, binomials, and trinomials,” “determine the degree of polynomials,” “add and subtract monomials,” “add and subtract polynomials,” and “evaluate a polynomial for a given value.” The rest of the cells are blank.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Questions & Answers

Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned $1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
Washing his dad’s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Gagan Reply
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
1,75 hrs is correct Mike
Emund
How did you come up with the answer?
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Marie Reply
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have?
Kendra Reply
Tickets to a Broadway show cost $35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were $47,150. How many adult and how many child tickets were sold?
dana Reply
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year? How do I do this
Tanesia Reply
how to square
Fiona Reply
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
Sabee Reply
Practice Key Terms 8

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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