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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

write in this form a/b answer should be in the simplest form 5%
August Reply
convert to decimal 9/11
August
Equation in the form of a pending point y+2=1/6(×-4)
Jose Reply
write in simplest form 3 4/2
August
definition of quadratic formula
Ahmed Reply
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
What Reply
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
Mckenzie Reply
x > $110,000
bruce
greater than $110,000
Michael
Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost $1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her $2.52 per pound, how many pounds of each berry should she use?
nawal Reply
$1.38 worth of strawberries + $1.14 worth of blueberries which= $2.52
Leitha
how
Zaione
is it right😊
Leitha
lol maybe
Robinson
8 pound of blueberries and 22 pounds of strawberries
Melissa
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound
Melissa
hmmmm...... ?
Robinson
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
The question asks how many pounds of each in order for her to have an average cost of $2.52. She needs 30 lb in all so 30 pounds times $2.52 equals $75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa
Vina has $4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have?
Mckenzie Reply
10 quarters 16 dimes 12 nickels
Leitha
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet.
Crispy Reply
wtf. is a tail wind or headwind?
Robert
48 miles per hour with headwind and 68 miles per hour with tailwind
Leitha
average speed is 58 mph
Leitha
Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150.
bruce
the jet will fly 9.68 hours to cover either distance
bruce
Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs $3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound?
Vonna Reply
33.336
Robinson
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot?
Imaan Reply
Ivan has $8.75 in nickels and quarters in his desk drawer. The number of nickels is twice the number of quarters. How many coins of each type does he have?
mikayla Reply
2q=n ((2q).05) + ((q).25) = 8.75 .1q + .25q = 8.75 .35q = 8.75 q = 25 quarters 2(q) 2 (25) = 50 nickles Answer check 25 x .25 = 6.25 50 x .05 = 2.50 6.25 + 2.50 = 8.75
Melissa
John has $175 in $5 and $10 bills in his drawer. The number of $5 bills is three times the number of $10 bills. How many of each are in the drawer?
mikayla Reply
7-$10 21-$5
Robert
Enrique borrowed $23,500 to buy a car. He pays his uncle 2% interest on the $4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total $23,500? (Round your answer to the nearest tenth of a percent.)
Parker Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hour longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Markeice Reply
8mph
michele
16mph
Robert
3.8 mph
Ped
16 goes into 80 5times while 20 goes into 80 4times and is 4mph faster
Robert
what is the answer for this 3×9+28÷4-8
What Reply
315
lashonna
how do you do xsquard+7x+10=0
What
(x + 2)(x + 5), then set each factor to zero and solve for x. so, x = -2 and x = -5.
bruce
I skipped it
What
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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