How to Add and Subtract Polynomials

8/11/2019 How to Add and Subtract Polynomials

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How to Add and Subtract PolynomialsStep By Step

worksheet on adding and subtracting polynomials ( 25 question pdf withanswer key)Related : Polynomial Home | Degree of Polynomial

Whether you want to add polynomials or subtract them, you follow a similar set ofsteps. Adding Polynomials Step 1) Arrange the Polynomial in standard form Standard form of a polynomial just means that the term with highest degree is firstand each of the following termsStep 2) Arrange the like terms in columns and add the like terms Example 1 Let's find the sum of the following two polynomials(3y 5 2y + y 4 + 2y 3 + 5) and (2y 5 + 3y 3 + 2+ 7)

In English, the tenses are past perfect, indicating that action was completed in the past ("I had gone to the store")simple past, indicating that action was occurring in the past ("I went to the store"); or, in the subjunctivemood, suggesting unreality ("Were I you...")

present perfect, indicating that action was recently completed ("I have gone to the store")simple present, indicating that action is occurring now ("I hate you"); or, in the subjunctive mood,suggesting greater possibility ("Be he alive or be he dead...")

future perfect, indicating that action will be completed in the future ("I will have gone to the store")simple future, indicating that action will occur in the future ("You will be mine")The FUTURE TENSE indicates that an action is in the future relative to the speaker or writer. There areno inflected forms for the future in English (nothing like those -ed or -s endings in the other tenses).Instead, the future tense employs the helping verbs will or shall with the base form of the verb:She will leave soon.We shall overcome.The future is also formed with the use of a form of "go" plus the infinitive of the verb:He is going to faint.
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He was sleeping all night long.

Were you sleeping all night long?

I wasn't resting , I was working !

Last night at nine o'clock John was washing the dishes.

Why was John washing the dishes?

Last week we were painting the house.

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TOEFL | English Grammar for Students | Forms Of The Tenses P

Forms Of The Tenses The forms of a tense may be expressed in different ways. There are three forms of tenses , the common, the progressive, and the emphat
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Practice Test

Common Form:

1. The earth moves.

2. The boy studied.

The common form of tenses is expressed in the above sentences .

Progressive Form:

1. I am writing.

2. I was reading when you came in.

The progressive form of tenses is expressed in the above sentences, and represents an act or state as continuing at the time referred to.

Emphatic Form:

1. I do work.

2. I did read an entire hour.

The emphatic form of tenses is expressed by the above sentences, and expresses an act or state with positiveness. Do or did help to make the eforms of tenses.

1. I did not see you yesterday.

2. He did not arrive as we expected.

Do and did do not render negative propositions emphatic, but simply make them more exact by inclosing the negative adverb withinverb .

1. Did you receive the book that I sent?

2. Know you the people?

3. Have you spoken with the gentleman?

We see from the above sentences that a sentence may be made interrogative by placing the verb or some part of it before the subject

Give the tenses of the verbs in the following sentences:

1. The governor is in town to-day, but he will leave to-night at eight o'clock.

2. You are now where you stood fifty years ago this very hour, with your brothers, and your neighbors, shoulder to shoulder, in the strife for ycountry.

3. Those who survived that day, and whose lives have been prolonged to the present hour, are now around you.

4. Some of them you have known in the trying scenes of war.

5. And when you shall here have exchanged your embraces, when you shall have once more pressed the hands which have been so often extegive succor in adversity, then look abroad upon this lovely land which your young valor defended.
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6. He had now entered the outskirts of the village.

7. There were rows of houses which he had never seen before, and those which had been his familiar haunts had disappeared.

8. Then all the people looked and saw that what the deep-sighted poet said was true.

9. I hope you will enjoy your vacation.

10. We are reading Emerson's "Behavior," and find that it will be all or more than we can master.

11. I do believe you, my boy; for you have always shown an honest, manly spirit.

12. Have the times so changed, that merit cannot win without influence?


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The FUTURE TENSE indicates that an action is in the future relative to the speaker orwriter. There are no inflected forms for the future in English (nothing like those -ed or -s endings in the other tenses). Instead, the future tense employs the helping verbs will or shall with the base form of the verb:

She will leave soon. We shall overcome.

The future is also formed with the use of a form of "go" plus the infinitive of the verb:

He is going to faint.

English can even use the present to suggest the future tense:

I am leaving later today."

9 Forms of the Past Tense By Mark Nichol


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Multiple variations of past tense that employ regular verbs occur in English. Explanations of thedistinctions follow. Note that each section includes examples of positive-declarative, negative-declarative, and interrogative forms.

1. Simple Past

A sentence in the simple-past form describes an event that occurred in the past:

They agreed with us.

They did not agree with us.

Did they agree with us?

Notice that in the first sentence, the verb form of agree is in past tense, but in the other examples,did does the heavy lifting of indicating the tense, so agree remains in present tense. In almost all

other variations of past tense, the form of the verb to be and the participle retain the same formregardless of the type of sentence.

2. Past Progressive (or Past Continuous)

Past-progressive statements and questions describe something that began in the past andcontinued to occur for a time before stopping:

They were agreeing with us.

They were not agreeing with us.

Were they agreeing with us?

3. Past Perfect

This tense form applies to events that began at a time preceding a period in the past:

They had agreed with us.

They had not agreed with us.

Had they agreed with us?

4. Past Perfect Progressive (or Past Perfect Continuous)

Sentences with this tense form describe something that occurred in the past and continued tooccur after the fact but in the present is no longer occurring:

They had been agreeing with us.


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They had not been agreeing with us.

Had they been agreeing with us?

5. Past Habitual

A sentence written in past-habitual tense describes an occurrence that once occurredcontinuously or repeatedly:

They used to agree with us.

They used to not agree with us.

(This formal usage is awkward and seldom used; we are more likely to write, They used todisagree with us. An informal version of the sentence, more likely to be used if no directantonym like d isagree is available for a given sentence, is They didnt use to agree with us.)

Used they to agree with us?

(This formal usage is rare. The informal form, much more common, is, Did they use to agreewith us?)

6. Time-Specific Past Habitual

A variation of the past-habitual tense includes a specific time frame:

Before, they would agree with us.

Before, they would not agree with us.

Before, would they agree with us?

7. Past Intensive

A sentence in the past-intensive form describes something confirmed as having occurred:

They did agree with us.

They did not agree with us.

Did they agree with us?

8. Future in the Past

A future-in-the-past construction describes something that was supposed to have occurred after atime in the past:


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They were going to agree with us.

They were not going to agree with us.

Were they going to agree with us?

Past Subjunctive This form is not numbered, because it is not, despite its name, a type of past tense, but it isidentified here to make that point. A sentence formed in the past subjunctive describes acounterfactual event:

If they were going to agree with us, they would have told us by now.

If they were not going to agree with us, they would have told us by now.

If they were they going to agree with us, would they have told us by now?

9. Past-Perfect Subjunctive

Sentences with this subjunctive form, by contrast, do have a past-tense sense:

Had they agreed with us, they would have told us by now.

Had they not agreed with us, they would have told us by now.

Had they agreed with us, would they have told us by now?

Anna Smoak Short Range Lesson Plan

Title of Lesson: Adding and Subtracting Polynomials Subject: Algebra IGrade level: 9 th

Teacher: Ms. Burgess

Objective(s): Students will be able to identify an expression as a monomial, binomial, or trinomial Students will be able to explain the difference between a monomial, binomial, and trinomial Students will be able to add polynomials Students will be able to subtract polynomials


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SCSDE Curriculum Standard(s) Addressed:EA-2.5 Carry out a procedure using the properties of real numbers (including commutative,

associative, and distributive) to simplify expressions.

EA-2.7 Carry out a procedure (including addition, subtraction, multiplication, and division by amonomial) to simplify polynomial expressions.

NCTM National Curriculum Standard(s) Addressed:Algebra: Represent and analyze mathematical situations and structures using algebraic symbols (grades9-12)

Use symbolic algebra to represent and explain mathematical relationships Understand the meaning of equivalent forms of expressions, equations, inequalities, and

relations

Communication: Organize and consolidate their mathematical thinking through communication

Communicate their mathematical thinking coherently and clearly to peers, teachers, andothers

Prerequisites: Students must be able to recognize as well as add and subtract monomials Students must be able to distribute a constant through a polynomial Students must be able to add, subtract, and multiply real numbers Students are able to use the process of combining like terms to simplify an expression

Materials/Preparation: Smartboard and whiteboard will be used

Powerpoint or Smart presentation will be minimized and ready for immediate use Students must have notebooks and pencils

Procedures: Introductory Activity (5 minutes):

o Students will complete a Frayer Model in their notebooks on the topic of monomials inwhich they will define the term (in their own words), list facts and/or characteristics,provide examples, and generate non-examples

Responses will then be shared with the classo Students will be asked what the root word mono- meanso Students will then be asked what the root word bi- means and will as a class generate as

many words that have the root bi- (such as bicycle) as possible. They will then be askedto predict what a binomial would be by looking at its root word. An example and a non-example of a binomial will be shown

o Students will then be asked what the root word tri-means and as a class will generate asmany words that have the root tri- (such as tricycle) as possible. They will then be askedto predict what a trinomial would be by looking at its root word

An example and non-example of a trinomial will be shown


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o Students will be asked to explain why xyz is a monomial while x + y + z is a trinomial.(I will have them take several minutes to write down a response to this problem in theirnotes and we will then have a whole class discussion.)

Main Activity (20 minutes):o Students will then be asked How can we represent x2 + 2x + 2 using Algebra Tiles? and

How can we represent 2x2 + 2x using Algebra Tiles? (provided) Both of which have strictly positive coefficients

o Students will be asked to Represent the sum of x2 + 2x + 2 and 2x 2 + 2x using AlgebraTiles.

o Students will then use this representation to simplify the expression using the process ofcombining like terms, previously learned

We will pause after collecting like terms to translate our modified pictorialrepresentation into a mathematical statement

Again after simplifying we will translate our visual representation of theexpression into a mathematical statement

o Students will then be asked How can we represent - x 2 + x + 5 using Algebra Tiles? andHow can we represent x2 + 2x - 2 using Algebra Tiles?

o Students will then have students represent the sum of - x 2 + x + 5 and x 2 + 2x - 2 usingAlgebra Tiles

Now students must account for a negative coefficient and a negative constantbut the two polynomials are still being added together

o Students will use the process of combining like terms to add the two polynomialstogether and simplify the expression

We will pause after collecting like terms to translate our modified pictorialrepresentation into a mathematical statement

Again after simplifying we will translate our visual representation of theexpression into a mathematical statement

Students will then be asked to find the sum of x 2 + 2x + 3 and 3x + 1 algebraically

without the use of Algebra Tiles. I will ask students to try this individually andthen check their work with a partner. After several minutes I will ask a pair topresent their strategy on the board for the rest of the class. I will then ask foralternate solution strategies.

Students will be asked to state the property that allows the removal ofparentheses around the polynomials

o I will then ask What is my first step in simplifying the expression (x + 1)?(students have already learned to distribute constant terms throughout an expressionhowever, I think it would be beneficial to start with this question because manystudents are having trouble with positive and negative signs during distribution)

Students will be asked to name the property (distribution) we use to simplifythe expression

o Students will then be asked to model the expression - (x 2 + x + 5) using Algebra Tiles I will ask for strategies for how to approach this problem. They will be asked Is

there anything that we have to do before we can use the Algebra Tiles? Afterdistributing the negative through the expression students will then explain thenecessary Algebra Tiles required to model the expression

o They will then be asked How can we represent (-2x2 + x - 3) using Algebra Tiles?


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o Students will then be asked How can we find the difference of the two expressions?They must realize that the subtraction sign must first be distributed to each term in thesecond expression.

o Students will be given the opportunity to use Algebra Tiles to represent this problemand use the process of combining like terms to simplify. They will translate theirpictorial representation to a mathematical expression after each step.

o Students will then be asked to simplify the problem (3x 2 5x + 3) (2x2 x 4) I will solve this problem along with the students on the board by implementing a

t-chart. Our mathematical steps will be shown in the left column and ourexplanation of our steps will be shown in the right column.

o Student s will then be presented with the problem Find the difference of ( 2x3 + 5x2 4x + 8) and ( 2x3 + 3x 4)

o As an extension problem students will be asked to Write an expression for theperimeter of the rectangle. Then simplify the expression. A vis ual of the rectangle willbe shown. Students will be asked to provide the formula for finding the perimeter of arectangle. I will then ask students how we might set this problem up.

Closure (10 minutes):o Students will complete a ticket out of the door: When two binomials are added, will

the sum always, sometimes, or never be a binomial? Explain your answer and giveexamples to support your answer.

Assessment: Students can successfully complete written in class and homework problems (included) which

require students to add and subtract polynomials with both positive and negative signs as wellas classify expressions as monomials, binomials, or trinomials

Students are able to answer the questions presented in the lesson as well as provide justification for their responses

Students are able to generalize their knowledge to answer and justify the ticket out of the

door activity

Adaptations: Students have the opportunity to come in during their AE period or tutoring after school for

extra assistance I have also increased the size of the font on the PowerPoint because several students in class

have difficulty seeing the Smartboard Optional guided practice worksheets will be given to all students that want one

Follow-up Lessons/Activities: The beginning of the next lesson will include a review of adding and subtracting polynomials

Students will complete homework problems for extra practice and will be given the opportunityto ask questions when they are reviewed the next day

How to Add and Subtract Polynomials

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