Adding and Subtracting Polynomials Polynomials in the …tristanbates.wikispaces.com/file/view/Alg.+I+-+Unit+4... · 2013-12-05 · Adding and Subtracting Polynomials Polynomials

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Unit 4 • Exponents, Radicals, and Polynomials 231

ACTIVITY

4.4Adding and Subtracting PolynomialsPolynomials in the SunSUGGESTED LEARNING STRATEGIES: Shared Reading, Questioning the Text, Create Representations, Vocabulary Organizer, Note Taking

A solar panel is a device that collects and converts solar energy into electricity or heat. ! e solar panel consists of interconnected solar cells. ! e panels can have di" ering numbers of solar cells and can come in square or rectangular shapes.

1. How many solar cells are in the panel below?

2. If a solar panel has four rows as the picture does, but can be extended to have an unknown number of columns, x, write an expression to give the number of solar cells that could be in the panel.

3. If a solar panel could have x rows and x columns of solar cells, write an expression that would give the total number of cells in the panel.

4. If you had 5 panels like those found in Item 3, write an expression that would give the total number of solar cells.

All the answers in Items 1–4 are called terms. A term is a number, variable or the product of a number and variable(s).

5. Write the sum of your answers from Items 1, 2, and 4.

CONNECT TO SCIENCESCIENCE

Solar panels, also known as photovoltaic panels, are made of semiconductor materials. A panel has both positive and negative layers of semiconductor material. When sunlight hits the semiconductor, electrons travel across the intersection of the two different layers of materials, creating an electric current.

ACADEMIC VOCABULARY

term

231-240_SB_A1_4-4_SE.indd 231231-240_SB_A1_4-4_SE.indd 231 1/6/10 4:51:18 PM1/6/10 4:51:18 PM

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232 SpringBoard® Mathematics with MeaningTM Algebra 1

Adding and Subtracting Polynomials ACTIVITY 4.4continued Polynomials in the SunPolynomials in the Sun

Expressions like the one in Item 5 are called polynomials. A polynomial is a single term or the sum of two or more terms.

6. List the terms of the polynomial you found in Item 5.

7. What are the coe! cients and constant terms of the polynomial in Item 5?

! e degree of a term is the sum of the exponents on the variables contained in the term.

8. Find the degree and coe" cient of each term in the polynomial 4x 5 + 12x 3 + x 2 - x + 5.

Term Degree Coe! cient4x 5 5

12x 3 12x 2

-x5

9. For the polynomial 2x 3y - 6x 2y 2 + 9xy - 13y 5 + 5x + 15, list each term and identify its degree and coe" cient.

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Vocabulary Organizer, Note Taking, Group Discussion

A constant term is a term that contains only a number, such as your answer to Item 1. The constant term of a polynomial is a term of degree zero. For example, 4 can also be written as 4x0

MATH TERMS

ACADEMIC VOCABULARY

A coeffi cient is the numeric factor of a term.

ACADEMIC VOCABULARY

A polynomial is a term or the sum of two or more terms.


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ACTIVITY 4.4continued

Adding and Subtracting PolynomialsPolynomials in the SunPolynomials in the Sun

SUGGESTED LEARNING STRATEGIES: Note Taking, Vocabulary Organizer, Interactive Word Wall

! e degree of a polynomial is the largest degree of any term in the polynomial.

10. Find the degree and constant term of each polynomial.

PolynomialDegree of

Polynomial Constant Term2x 2 + 3x + 7 2

-5y 3 + 4y 2 - 8y - 336 + 12x + x 2 36

! e standard form of a polynomial is a polynomial written in descending order of degree. ! e leading coe! cient is the coe" cient of a polynomial’s leading term when it is written in standard form.

ACADEMIC VOCABULARY

degree of a polynomial

READING MATH

The prefi xes mono (one), bi (two), tri (three) and poly (many) appear in many math terms such as bisect (cut in half ), triangle (three-sided fi gure), polygon (many-sided) fi gure.

Descending order of degree means that the term that has the highest degree is written fi rst, the term with the next highest degree is written next, and so on.

MATH TERMS

A polynomial can be classi# ed by the number of terms it has when it is in simplest form.

Name Number of Terms n Examplesmonomial 1 8 or –2x or 3x 2

binomial 2 3x + 2 or 4x 2 - 7xtrinomial 3 -x 2 - 3x + 9

polynomial n > 3 9x 4 - x 3 - 3x 2 + 7x - 2

Degree of the polynomial

Leading coe" cient

Term

Constant7 x 5 + 2 x 2 - 3


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SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Self/Peer Revision

11. Fill in the missing information in the table below.

12. If a square solar panel with an unknown number of cells along the edge can be represented by x 2, how many cells would be in one column of the panel?

A square solar panel with x rows and x columns can be represented by the algebra tile:

A column of x cells can be represented by using the tile

x

, and a single solar cell can be represented by +1 .

Suppose there were 3 square solar panels that each had x columns and x rows, 2 columns with x cells, and 3 single solar cells. You can represent 3x 2 + 2x + 3 using algebra tiles.

x2 x2

+1

+1

+1

x2 x x

PolynomialNumber of Terms Name

Leading coe! cient

Constant term Degree

3x 2 - 5x-2x 2 + 13x + 6

15x 2 5p 3 + 2p 2 - p - 7

a 2 - 25 0.23x 3 + 0.54x 2 -

0.58x + 0.0218-9.8t 2 - 20t + 150

x2


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SUGGESTED LEARNING STRATEGIES: Group Discussion

13. Represent 2x 2 - 3x + 2 using algebra tiles. Draw a picture of the representation below.

The additive inverse of the x 2, x, and 1 algebra tiles can be repre-sented with another color, or the fl ip side of the tile.

Adding PolynomialsAdding polynomials using algebra tiles can be done by:

modeling each polynomial identifying and removing zero pairs writing the new polynomial

EXAMPLE 1

Add (3x 2 - 3x - 5) + (2x 2 + 5x + 3) using algebra tiles.

Step 1: Model the polynomials. 3x 2 – 3x – 5 2x 2 + 5x + 3

x2 x2 x2

–1

–1

–1

–1

–1 x2 x2

+1

+1

+1

–x –x –x x x x x x

Step 2: Identify and remove zero pairs.

x2 x2 x2 x2 x2

–1

–1

–1

–1

–1

+1

+1

+1

x x –x x –x x –x x

Step 3: Combine like tiles.

x2 x2 x2x2 x2

–1

–1x x

Step 4: Write the polynomial for the model in Step 3. 5x 2 + 2x - 2Solution: (3x 2 - 3x - 5) + (2x 2 + 5x + 3) = 5x 2 + 2x - 2


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SUGGESTED LEARNING STRATEGIES: Create Representations, Use Manipulatives

TRY THESE A

Add using algebra tiles.a. (x 2 - 2) + (2x 2 + 5)

b. (2y 2 + 3y + 6) + (3y 2 - 4)

c. (2x 2 + 3x + 9) + (-x 2 - 4x - 6)

d. (5 - 3x + x 2 ) + (2x + 4 - 3x 2)

14. Can you use algebra tiles to add (4x 4 + 3x 2 + 15) + (x 4 + 10x3 - 4x 2 + 22x - 23)? If so, model the polynomials and add. If not, explain why.

Like terms in an expression are the terms that have the same variable and exponent for that variable. All constants are like terms.

15. State whether the terms are like or unlike terms. Explain.

a. 2x; 2x 3

b. 5; 5x

c. –3y; 3y

d. x 2y; xy 2

e. 14; –0.6

16. Using vocabulary from this unit, describe a method that could be used to add polynomials without using algebra tiles.


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SUGGESTED LEARNING STRATEGIES: Note Taking

! e method you described in Item 16 can be used to add polynomials algebraically.

EXAMPLE 2

Add (3x 3 + 2x 2 - 5x + 7) + (4x 2 + 2x - 3) horizontally and vertically.

Horizontally

Step 1: Find like terms. (3x 3 + 2x 2 - 5x + 7) + (4x 2 + 2x - 3)

Step 2: Group like terms. = (3x 3) + (2x 2 + 4x 2) + (-5x + 2x) +(7 - 3)

Step 3: Add the coe! cients of like terms. = 3x 3 + 6x 2 - 3x + 4Solution: (3x 3 + 2x 2 - 5x + 7) + (4x 2 + 2x - 3) = 3x 3 + 6x 2 - 3x + 4

Vertically

Step 1: Vertically align the like terms. 3x 3 + 2x 2 - 5x + 7Step 2: Add the coe! cients of like + 4x 2 + 2x - 3 terms. 3x 3 + 6x 2 - 3x + 4Solution: (3x 3 + 2x 2 - 5x + 7) + (4x 2 + 2x - 3) = 3x 3 + 6x 2 - 3x + 4

TRY THESE B

Add using either the horizontal or the vertical method.

a. (4x 2 + 3) + (x 2 - 3x + 5)

b. (10y 2 + 8y + 6) + (17y 2 - 11)

c. (9x 2 + 15x + 21) + (-13x 2 - 11x - 26)

d. (18 + 21x 2) + (3x + 4 - 52x 2)

Recall that the commutative and associative properties of addition allow you to re-order and group like terms.


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SUGGESTED LEARNING STRATEGIES: Create Representations, Use Manipulatives

Subtracting PolynomialsSubtracting polynomials using algebra tiles can be done by

modeling the ! rst polynomialmodeling the additive inverse of the second polynomialidentifying and removing zero pairswriting the new polynomial

EXAMPLE 3

Subtract (2x 2 + x - 3) - (x 2 + 4x + 1) using algebra tiles.

Step 1: Model 2x 2 + x - 3 and the additive inverse of x 2 + 4x + 1.

x2 x2 –x2

–1

–1

–1

–1

x –x –x –x –x

Step 2: Identify and remove zero pairs.

x2

–1 –1

–1

–1

–x –x –x

x2 –x2 –x x

Step 3: Combine like tiles.

x2

–1 –1

–1 –1–x –x –x

Step 4: Write the polynomial for the model in Step 4. x 2 - 3x - 4Solution: (2x 2 + x - 3) - (x 2 + 4x + 1) = x 2 - 3x - 4

TRY THESE C

a. Use algebra tiles to represent the additive inverse of 3x 2 - 2x - 4. " en draw the representation.

Remember that subtraction can be represented by adding the opposite, the additive inverse.


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SUGGESTED LEARNING STRATEGIES: Note Taking

TRY THESE C (continued )

Subtract using algebra tiles.

b. (3x 2 - 1) - (2x 2 + 6)

c. (3y 2 + 4y + 7) - (2y 2 - 2)

d. (x 2 + x + 2) - (-2x 2 - 5x - 7)

e. (4 + 2x + 4x 2) - (2x + 4 - 3x 2)

To subtract a polynomial you add its opposite, or subtract each of its terms.

EXAMPLE 4

Subtract (2x 3 + 8x 2 + x + 10) - (5x 2 - 4x + 6) horizontally and vertically.

Horizontally

Step 1: Distribute the (2x 3 + 8x 2 + x + 10) - (5x 2 - 4x + 6) negative.

Step 2: Find like terms. = 2x 3 + 8x 2 + x + 10 - 5x 2 + 4x - 6

Step 3: Group like terms = 2x3 + (8x2 – 5x2) + (x + 4x) + (10 – 6)

Step 4: Combine = 2x3 + 3x2 + 5x + 4 coe! cients of like terms.

Solution: (2x 3 + 8x 2 + x + 10) - (5x 2 - 4x + 6) = 2x 3 + 3x 2 + 5x + 4

Vertically

Step 1: Vertically align the like terms. 2x 3 + 8x 2 + x + 10 -(5x 2 - 4x + 6)

2x 3 + 8x 2 + x + 10Step 2: Distribute the negative. -5x 2 + 4x - 6Step 3: Combine coe! cients of like terms. 2x 3 + 3x 2 + 5x + 4Solution: (2x 3 + 8x 2 + x + 10) - (5x 2 - 4x + 6) = 2x 3 + 3x 2 + 5x + 4


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SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Note Taking

To multiply a monomial by a polynomial, use the distributive property. Multiply each term of the trinomial by the monomial.

TRY THESE D

Subtract using either the horizontal or the vertical method.

a. (5x 2 - 5) - (x 2 + 7)

b. (2x 2 + 3x + 2) - (-5x 2 - 2x - 9)

c. (y 2 + 3y + 8) - (4y 2 - 9)

d. (12 + 5x + 14x 2) - (8x + 15 - 7x 2)

17. Suppose there are 10 solar panels that have 3x 2 + 7x + 3 cells on each panel. Write a polynomial that represents the total number of solar cells in all 10 panels combined.

18. Based on your work in this activity, do you think that the set of polynomials is closed under the operations of addition and subtraction? Explain.

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

For Items 1–5, use the polynomial 4x 3 + 3x 2 - 9x + 7.

1. Name the coe! cients of the polynomial

2. List the terms, and specify the degree of each term.

3. What is the degree of the polynomial?

4. What is the leading coe! cient of the polynomial?

5. What is the constant term of the polynomial?

Find the sum or di! erence.

6. (3x 2 + 2x + 7) + (-2x 3 + 3x 2 - 8x - 6)

7. (2x 3 + 4x - 9) - (-7x 3 + 2x 2 - 5x + 13)

8. MATHEMATICAL R E F L E C T I O N

Describe the similarities and di" erences in adding

and subtracting polynomials. What method of adding polynomials do you feel most comfortable with?

231-240_SB_A1_4-4_SE.indd 240231-240_SB_A1_4-4_SE.indd 240 1/20/11 11:16:05 AM1/20/11 11:16:05 AM

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Adding and Subtracting Polynomials Polynomials in the …tristanbates.wikispaces.com/file/view/Alg.+I+-+Unit+4... · 2013-12-05 · Adding and Subtracting Polynomials Polynomials