Algebra unit 8.1

UNIT 8.1 ADDING AND SUBTRACTINGUNIT 8.1 ADDING AND SUBTRACTINGPOLYNOMIALSPOLYNOMIALS

Warm UpSimplify each expression by combining like terms.

1. 4x + 2x

2. 3y + 7y

3. 8p – 5p

4. 5n + 6n2

Simplify each expression.

5. 3(x + 4)

6. –2(t + 3)

7. –1(x2 – 4x – 6)

6x

10y

3p

not like terms

3x + 12

–2t – 6

–x2 + 4x + 6

Add and subtract polynomials.

Objective

Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.

Add or Subtract..

Example 1: Adding and Subtracting Monomials

A. 12p3 + 11p2 + 8p3

12p3 + 11p2 + 8p3

12p3 + 8p3 + 11p2

20p3 + 11p2

Identify like terms.Rearrange terms so that like

terms are together.Combine like terms.

B. 5x2 – 6 – 3x + 8

5x2 – 6 – 3x + 8

5x2 – 3x + 8 – 6

5x2 – 3x + 2

Identify like terms.Rearrange terms so that like

terms are together.Combine like terms.

Add or Subtract..

Example 1: Adding and Subtracting Monomials

C. t2 + 2s2 – 4t2 – s2

t2 – 4t2 + 2s2 – s2 t2 + 2s2 – 4t2 – s2

–3t2 + s2

Identify like terms.Rearrange terms so that

like terms are together.Combine like terms.

D. 10m2n + 4m2n – 8m2n

10m2n + 4m2n – 8m2n

6m2n

Identify like terms.

Combine like terms.

Like terms are constants or terms with the same variable(s) raised to the same power(s). To review combining like terms, see lesson 1-7.

Remember!

Check It Out! Example 1

a. 2s2 + 3s2 + s

Add or subtract.

2s2 + 3s2 + s

5s2 + s

b. 4z4 – 8 + 16z4 + 24z4 – 8 + 16z4 + 2

4z4 + 16z4 – 8 + 220z4 – 6


Combine like terms.




c. 2x8 + 7y8 – x8 – y8

Add or subtract.

2x8 + 7y8 – x8 – y8 2x8 – x8 + 7y8 – y8 x8 + 6y8

d. 9b3c2 + 5b3c2 – 13b3c2

9b3c2 + 5b3c2 – 13b3c2 b3c2


Combine like terms.



Polynomials can be added in either vertical or horizontal form.

In vertical form, align the like terms and add:

In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms.

(5x2 + 4x + 1) + (2x2 + 5x + 2)

= (5x2 + 2x2 + 1) + (4x + 5x) + (1 + 2)

= 7x2 + 9x + 3

5x2 + 4x + 1+ 2x2 + 5x + 2

7x2 + 9x + 3

Add.Example 2: Adding Polynomials

A. (4m2 + 5) + (m2 – m + 6)

(4m2 + 5) + (m2 – m + 6)

(4m2 + m2) + (–m) +(5 + 6)

5m2 – m + 11


Group like terms together.

Combine like terms.

B. (10xy + x) + (–3xy + y)

(10xy + x) + (–3xy + y)

(10xy – 3xy) + x + y

7xy + x + y



Combine like terms.

Add.Example 2C: Adding Polynomials

(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)


Group like terms together within each polynomial.

Combine like terms.

(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)

(6x2 + 3x2 – 8x2) + (3y – 4y – 2y)

Use the vertical method. 6x2 – 4y+ –5x2 + y

x2 – 3y Simplify.

Add.Example 2D: Adding Polynomials



Combine like terms.


Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a).

(5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a)

(5a3 + 7a3) + (3a2 + 12a2) + (–10a – 6a)

12a3 + 15a2 – 16a



Combine like terms.

To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial:

–(2x3 – 3x + 7)= –2x3 + 3x – 7

Subtract.

Example 3A: Subtracting Polynomials

(x3 + 4y) – (2x3)

(x3 + 4y) + (–2x3)

(x3 + 4y) + (–2x3)

(x3 – 2x3) + 4y

–x3 + 4y

Rewrite subtraction as addition of the opposite.



Combine like terms.

Subtract.

Example 3B: Subtracting Polynomials

(7m4 – 2m2) – (5m4 – 5m2 + 8)

(7m4 – 2m2) + (–5m4 + 5m2 – 8)

(7m4 – 5m4) + (–2m2 + 5m2) – 8

(7m4 – 2m2) + (–5m4 + 5m2 – 8)

2m4 + 3m2 – 8




Combine like terms.

Subtract.

Example 3C: Subtracting Polynomials

(–10x2 – 3x + 7) – (x2 – 9)

(–10x2 – 3x + 7) + (–x2 + 9)

(–10x2 – 3x + 7) + (–x2 + 9)

–10x2 – 3x + 7–x2 + 0x + 9

–11x2 – 3x + 16



Use the vertical method.Write 0x as a placeholder.Combine like terms.

Subtract.

Example 3D: Subtracting Polynomials

(9q2 – 3q) – (q2 – 5)

(9q2 – 3q) + (–q2 + 5)

(9q2 – 3q) + (–q2 + 5)

9q2 – 3q + 0+ − q2 – 0q + 5

8q2 – 3q + 5


Identify like terms.Use the vertical method.Write 0 and 0q as

placeholders.

Combine like terms.


Subtract.

(2x2 – 3x2 + 1) – (x2 + x + 1)

(2x2 – 3x2 + 1) + (–x2 – x – 1)

(2x2 – 3x2 + 1) + (–x2 – x – 1)

–x2 + 0x + 1 + –x2 – x – 1

–2x2 – x



Use the vertical method.Write 0x as a placeholder.

Combine like terms.

A farmer must add the areas of two plots of land to determine the amount of seed to plant. The area of plot A can be represented by 3x2 + 7x – 5 and the area of plot B can be represented by 5x2 – 4x + 11. Write a polynomial that represents the total area of both plots of land.

Example 4: Application

(3x2 + 7x – 5)(5x2 – 4x + 11)8x2 + 3x + 6

Plot A.Plot B.

Combine like terms.

+


The profits of two different manufacturing plants can be modeled as shown, where x is the number of units produced at each plant.

Use the information above to write a polynomial that represents the total profits from both plants.

–0.03x2 + 25x – 1500 Eastern plant profit.

–0.02x2 + 21x – 1700 Southern plant profit.Combine like terms.

+–0.05x2 + 46x – 3200

Lesson Quiz: Part I

Add or subtract.

1. 7m2 + 3m + 4m2

2. (r2 + s2) – (5r2 + 4s2)

3. (10pq + 3p) + (2pq – 5p + 6pq)

4. (14d2 – 8) + (6d2 – 2d +1)

(–4r2 – 3s2)

11m2 + 3m

18pq – 2p

20d2 – 2d – 7

5. (2.5ab + 14b) – (–1.5ab + 4b) 4ab + 10b

Lesson Quiz: Part II

6. A painter must add the areas of two walls to determine the amount of paint needed. The area of the first wall is modeled by 4x2 + 12x + 9, and the area of the second wall is modeled by

36x2 – 12x + 1. Write a polynomial that represents the total area of the two walls.

40x2 + 10

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Algebra unit 8.1

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Transcript of Algebra unit 8.1