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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

hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
where's the answers?
Ed Reply
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
What does 3 times your weight right now
Cherokee Reply
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Kendra Reply
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
Malik Reply
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
Adda Reply
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Adda
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
Adda
(×+1)(×-4) = x^2-3×-4
Gee
yeah
Asamoah
Quadratic equations involving factorization
Winner Reply
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
Please help me solve this 10x+14=-2x+38
Erica
the previous equation should be 3x = 1/6 x=1/18
Sriram
for the new one 10x + 2x = 38 - 14
Sriram
12x = 24 x=2
Sriram
10x + 14 = -2x +38 10x + 2x = 38 - 14 12x = 24 divide both sides by the coefficient of x, which is 12 therefore × = 2
vida
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon?
Mohamed Reply
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
muhammad Reply
really
Asamoah
4x-7y=8 2x-7y=1 what is the answer?
Ramil Reply
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
true
bismark
factoriz e
usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
Peggie
Ok cool answer peggie
Frank
thanks
Ramil
x=7/2 y=6/7
Asamoah
Thanks , all of you are correct.
Joseph
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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