6.1 Add and subtract polynomials (Page 2/12)

Page 2 / 12

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

This table has 11 rows and 5 columns. The first column is a header column, and it names each row. The first row is named “Monomial,” and each cell in this row contains a different monomial. The second row is named “Degree,” and each cell in this row contains the degree of the monomial above it. The degree of 14 is 0, the degree of 8y squared is 2, the degree of negative 9x cubed y to the fifth power is 8, and the degree of negative 13a is 1. The third row is named “Binomial,” and each cell in this row contains a different binomial. The fourth row is named “Degree of each term,” and each cell contains the degrees of the two terms in the binomial above it. The fifth row is named “Degree of polynomial,” and each cell contains the degree of the binomial as a whole.” The degrees of the terms in a plus 7 are 0 and 1, and the degree of the whole binomial is 1. The degrees of the terms in 4b squared minus 5b are 2 and 1, and the degree of the whole binomial is 2. The degrees of the terms in x squared y squared minus 16 are 4 and 0, and the degree of the whole binomial is 4. The degrees of the terms in 3n cubed minus 9n squared are 3 and 2, and the degree of the whole binomial is 3. The sixth row is named “Trinomial,” and each cell in this row contains a different trinomial. The seventh row is named “Degree of each term,” and each cell contains the degrees of the three terms in the trinomial above it. The eighth row is named “Degree of polynomial,” and each cell contains the degree of the trinomial as a whole. The degrees of the terms in x squared minus 7x plus 12 are 2, 1, and 0, and the degree of the whole trinomial is 2. The degrees of the terms in 9a squared plus 6ab plus b squared are 2, 2, and 2, and the degree of the trinomial as a whole is 2. The degrees of the terms in 6m to the fourth power minus m cubed n squared plus 8mn to the fifth power are 4, 5, and 6, and the degree of the whole trinomial is 6. The degrees of the terms in z to the fourth power plus 3z squared minus 1 are 4, 2, and 0, and the degree of the whole trinomial is 4. The ninth row is named “Polynomial,” and each cell contains a different polynomial. The tenth row is named “Degree of each term,” and the eleventh row is named “Degree of polynomial.” The degrees of the terms in b plus 1 are 1 and 0, and the degree of the whole polynomial is 1. The degrees of the terms in 4y squared minus 7y plus 2 are 2, 1, and 0, and the degree of the whole polynomial is 2. The degrees of the terms in 4x to the fourth power plus x cubed plus 8x squared minus 9x plus 1 are 4, 3, 2, 1, and 0, and the degree of the whole polynomial is 4.

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

Find the degree of the following polynomials.

ⓐ $10 y$
ⓑ $4 x^{3} - 7 x + 5$
ⓒ $−15$
ⓓ $−8 b^{2} + 9 b - 2$
ⓔ $8 x y^{2} + 2 y$

Solution

ⓐ
$\begin{matrix} 10 y \\ The exponent of y is one. y = y^{1} & The degree is 1. \end{matrix}$
ⓑ
$\begin{matrix} 4 x^{3} - 7 x + 5 \\ The highest degree of all the terms is 3. & The degree is 3. \end{matrix}$
ⓒ
$\begin{matrix} −15 \\ The degree of a constant is 0. & The degree is 0. \end{matrix}$
ⓓ
$\begin{matrix} −8 b^{2} + 9 b - 2 \\ The highest degree of all the terms is 2. & The degree is 2. \end{matrix}$
ⓔ
$\begin{matrix} 8 x y^{2} + 2 y \\ The highest degree of all the terms is 3. & The degree is 3. \end{matrix}$

Got questions? Get instant answers now!

Find the degree of the following polynomials:

ⓐ $−15 b$ ⓑ $10 z^{4} + 4 z^{2} - 5$ ⓒ $12 c^{5} d^{4} + 9 c^{3} d^{9} - 7$ ⓓ $3 x^{2} y - 4 x$ ⓔ $−9$

ⓐ $1$ ⓑ $4$ ⓒ $12$ ⓓ 3 ⓔ 0

Got questions? Get instant answers now!

Find the degree of the following polynomials:

ⓐ $52$ ⓑ $a^{4} b - 17 a^{4}$ ⓒ $5 x + 6 y + 2 z$ ⓓ $3 x^{2} - 5 x + 7$ ⓔ $− a^{3}$

ⓐ $0$ ⓑ $5$ ⓒ $1$ ⓓ 2 ⓔ 3

Got questions? Get instant answers now!

Add and subtract monomials

You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

Add: $25 y^{2} + 15 y^{2}$ .

Solution

$\begin{matrix} 25 y^{2} + 15 y^{2} \\ Combine like terms. & 40 y^{2} \end{matrix}$

Got questions? Get instant answers now!

Add: $12 q^{2} + 9 q^{2} .$

$21 q^{2}$

Got questions? Get instant answers now!

Add: $−15 c^{2} + 8 c^{2} .$

$−7 c^{2}$

Got questions? Get instant answers now!

Subtract: $16 p - (−7 p)$ .

Solution

$\begin{matrix} 16 p - (−7 p) \\ Combine like terms. & 23 p \end{matrix}$

Got questions? Get instant answers now!

Subtract: $8 m - (−5 m) .$

$13 m$

Got questions? Get instant answers now!

Subtract: $−15 z^{3} - (−5 z^{3}) .$

$−10 z^{3}$

Got questions? Get instant answers now!

Remember that like terms must have the same variables with the same exponents.

Simplify: $c^{2} + 7 d^{2} - 6 c^{2}$ .

Solution

$\begin{matrix} c^{2} + 7 d^{2} - 6 c^{2} \\ Combine like terms. & −5 c^{2} + 7 d^{2} \end{matrix}$

Got questions? Get instant answers now!

Add: $8 y^{2} + 3 z^{2} - 3 y^{2} .$

$5 y^{2} + 3 z^{2}$

Got questions? Get instant answers now!

Add: $3 m^{2} + n^{2} - 7 m^{2} .$

$−4 m^{2} + n^{2}$

Got questions? Get instant answers now!

Simplify: $u^{2} v + 5 u^{2} - 3 v^{2}$ .

Solution

$\begin{matrix} u^{2} v + 5 u^{2} - 3 v^{2} \\ There are no like terms to combine. & u^{2} v + 5 u^{2} - 3 v^{2} \end{matrix}$

Got questions? Get instant answers now!

Simplify: $m^{2} n^{2} - 8 m^{2} + 4 n^{2} .$

There are no like terms to combine.

Got questions? Get instant answers now!

Simplify: $p q^{2} - 6 p - 5 q^{2} .$

There are no like terms to combine.

Got questions? Get instant answers now!

Add and subtract polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Find the sum: $(5 y^{2} - 3 y + 15) + (3 y^{2} - 4 y - 11) .$

Solution

Identify like terms.
Rearrange to get the like terms together.
Combine like terms.

Got questions? Get instant answers now!

Find the sum: $(7 x^{2} - 4 x + 5) + (x^{2} - 7 x + 3) .$

$8 x^{2} - 11 x + 1$

Got questions? Get instant answers now!

Find the sum: $(14 y^{2} + 6 y - 4) + (3 y^{2} + 8 y + 5) .$

$17 y^{2} + 14 y + 1$

Got questions? Get instant answers now!

Find the difference: $(9 w^{2} - 7 w + 5) - (2 w^{2} - 4) .$

Solution


Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

Got questions? Get instant answers now!

Find the difference: $(8 x^{2} + 3 x - 19) - (7 x^{2} - 14) .$

$15 x^{2} + 3 x - 5$

Got questions? Get instant answers now!

Find the difference: $(9 b^{2} - 5 b - 4) - (3 b^{2} - 5 b - 7) .$

$6 b^{2} + 3$

Got questions? Get instant answers now!

Subtract: $(c^{2} - 4 c + 7)$ from $(7 c^{2} - 5 c + 3)$ .

Solution



Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

Got questions? Get instant answers now!

Subtract: $(5 z^{2} - 6 z - 2)$ from $(7 z^{2} + 6 z - 4)$ .

$2 z^{2} + 12 z - 2$

Got questions? Get instant answers now!

Subtract: $(x^{2} - 5 x - 8)$ from $(6 x^{2} + 9 x - 1)$ .

$5 x^{2} + 14 x + 7$

Got questions? Get instant answers now!

Find the sum: $(u^{2} - 6 u v + 5 v^{2}) + (3 u^{2} + 2 u v)$ .

Solution

$\begin{matrix} (u^{2} - 6 u v + 5 v^{2}) + (3 u^{2} + 2 u v) \\ Distribute. & u^{2} - 6 u v + 5 v^{2} + 3 u^{2} + 2 u v \\ Rearrange the terms, to put like terms together. & u^{2} + 3 u^{2} - 6 u v + 2 u v + 5 v^{2} \\ Combine like terms. & 4 u^{2} - 4 u v + 5 v^{2} \end{matrix}$

Got questions? Get instant answers now!

Find the sum: $(3 x^{2} - 4 x y + 5 y^{2}) + (2 x^{2} - x y)$ .

$5 x^{2} - 5 x y + 5 y^{2}$

Got questions? Get instant answers now!

Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.

John Reply

taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.

Faith

While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma

Kamara

which drugs can we use for ulcers

Ummi Reply

omeprazole

Kamara

what

Renee

what is this

Renee

is a drug

Kamara

of anti-ulcer

Kamara

Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit

Patrick

what is the function of lymphatic system

Nency Reply

Not really sure

Eli

to drain extracellular fluid all over the body.

asegid

The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of

asegid

to transport fluids fats proteins and lymphocytes to the blood stream as lymph

Adama

what is anatomy

Oyindarmola Reply

Anatomy is the identification and description of the structures of living things

Kamara

what's the difference between anatomy and physiology

Oyerinde Reply

Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.

AI-Robot

what is enzymes all about?

Mohammed Reply

Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems

Kamara

yes

Prince

how does the stomach protect itself from the damaging effects of HCl

Wulku Reply

little girl okay how does the stomach protect itself from the damaging effect of HCL

Wulku

it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL

Kamara

function of digestive system

Ali Reply

function of digestive

Ali

the diagram of the lungs

Adaeze Reply

what is the normal body temperature

Diya Reply

37 degrees selcius

Xolo

37°c

Stephanie

please why 37 degree selcius normal temperature

Mark

36.5

Simon

37°c

Iyogho

the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature

Stephanie

37A c

Wulku

what is anaemia

Diya Reply

anaemia is the decrease in RBC count hemoglobin count and PVC count

Eniola

what is the pH of the vagina

Diya Reply

how does Lysin attack pathogens

Diya

acid

Mary

I information on anatomy position and digestive system and there enzyme

Elisha Reply

anatomy of the female external genitalia

Muhammad Reply

Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)

Theophilus Reply

what's lochia albra

Kizito

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 8

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask

	19 AP 18 Cardiovascular System Blood Essay By OpenStax Start Flashcards
	16 AP 16 Neurological Essay Exam By OpenStax Start Flashcards
	11 Clin Sci I - GI Twedt quiz 2 By Brooke Delaney Start Exam
	15 Sociology 15 Religion MCQ By OpenStax Start Quiz
©flickr:	Anatomy Physiology By Jemekia Weeden Start Quiz
	Biology Exam Final By Savannah Parrish Start Exam
	Understanding basic music theory By OpenStax Read Online Course
	26 AP Key Terms 26 Fluid, Electrolyte, Acid-Base By OpenStax Start Key Terms
	7 Biology 07 Cellular Respiration MCQ By OpenStax Start Quiz
©flickr: Alexander	Mechanics I MCQ By Stephanie Redfern Start Quiz