Warm-upWarm-up

State whether each expression is a polynomial. If the State whether each expression is a polynomial. If the expession is a polynomial, identify it as either a expession is a polynomial, identify it as either a monomial, binomial, or trinomial and give its degree.monomial, binomial, or trinomial and give its degree.

1. 8a1. 8a22 + 5ab + 5ab

2. 3x2. 3x22 + 4x – 7/x + 4x – 7/x

3. 5x3. 5x22 + 7x + 2 + 7x + 2

4. 6a4. 6a22bb22 + 7ab + 7ab55 – 6b – 6b33

5. w5. w22 x - + 6x x - + 6x7w

3

Warm-upWarm-up

State whether each expression is a polynomial. If the State whether each expression is a polynomial. If the expession is a polynomial, identify it as either a expession is a polynomial, identify it as either a monomial, binomial, or trinomial and give its degree.monomial, binomial, or trinomial and give its degree.

1. 8a1. 8a22 + 5ab + 5ab binomial; 2binomial; 2

2. 3x2. 3x22 + 4x – 7/x + 4x – 7/x not a polynomial not the product not a polynomial not the product of # of # and variableand variable

3. 5x3. 5x22 + 7x + 2 + 7x + 2 trinomial; 2trinomial; 2

4. 6a4. 6a22bb22 + 7ab + 7ab55 – 6b – 6b33 trinomial; 2trinomial; 2

5. w5. w22 x - + 6x x - + 6x trinomial; 2trinomial; 27w

3

Homework 6.5Homework 6.5

1.1. 22

2.2. 33

3.3. 44

4.4. 11

5.5. 44

6.6. 44

7.7. 44

8.8. 88

9. 49. 4

10. 510. 5

11. 5 + 2x11. 5 + 2x22 + 3x + 3x33 + x + x44

12. 1 + 3x + 2x12. 1 + 3x + 2x22

13. - 6 + 5x + 3x13. - 6 + 5x + 3x22

14. 2 + x + 9x14. 2 + x + 9x22 + x + x33

15. - 3 + 4x – x15. - 3 + 4x – x22 + 3x + 3x33

16. – 2x + x16. – 2x + x22 - x - x33 + x + x44

Homework 6.5Homework 6.5

17. 6 + 12x + 6x17. 6 + 12x + 6x22 + x + x33

18. 21r18. 21r22 + 7r + 7r55x – rx – r22xx2 2 – 15x– 15x33

19. 5x19. 5x33 - 3x - 3x22 + x + x + 4 + 4

20. - x20. - x33 + x + x22 - x - x + 1 + 1

21. 3x21. 3x33 + x + x22 - x - x + 27 + 27

22. 3x22. 3x33 + x + x22 + x + x - 17 - 17

23. x23. x33 + x - 1 + x - 1

24. 3x24. 3x33 + x + x22 - x - x + 64 + 64

Homework 6.5Homework 6.5

25. - x25. - x33 + x + 25 + x + 25

26. 26. ⅓p⅓pxx33 + p + p33 x x22 + px + px + 5p + 5p

6.6 Adding and 6.6 Adding and Subtracting Subtracting PolynomialsPolynomials

CORD MathCORD Math

Mrs. SpitzMrs. Spitz

Fall 2006Fall 2006

Objectives:Objectives:

After studying this lesson, you After studying this lesson, you should be able to add and should be able to add and subtract polynomials.subtract polynomials.

Assignment:Assignment:

6.6 Worksheet6.6 Worksheet

Application:Application:

The standard The standard measurement for a window measurement for a window is the united inch. The is the united inch. The united inch measurement united inch measurement of a window is equal to the of a window is equal to the sum of the length of the sum of the length of the length and the width of the length and the width of the window. If the length of window. If the length of the window at the right is the window at the right is 2x + 8 and the width is x – 2x + 8 and the width is x – 3 inches, what is the size 3 inches, what is the size of the window in united of the window in united inches?inches?

x – 3 in.

Application:Application:

The size of the window is (2x The size of the window is (2x + 8) + (x – 3) inches. To add + 8) + (x – 3) inches. To add two polynomials, add the like two polynomials, add the like terms.terms.

= (2x +8) + (x - 3)= (2x +8) + (x - 3) = 2x + 8 + x – 3= 2x + 8 + x – 3 = (2x + x) + (8 – 3) = (2x + x) + (8 – 3) = 3x + 5= 3x + 5The size of the window in The size of the window in

united inches is 3x + 5 united inches is 3x + 5 inches. inches.

x – 3 in.

ApplicationApplication

You can add polynomials by You can add polynomials by grouping the like terms together grouping the like terms together and then finding the sum (as in and then finding the sum (as in the example previous), or by the example previous), or by writing them in column form.writing them in column form.

Example 1: Find (3yExample 1: Find (3y22 + 5y – 6) + 5y – 6) + (7y+ (7y22 -9) -9)

Method 1: Group the like terms Method 1: Group the like terms together.together.(3y(3y22 + 5y – 6) + (7y + 5y – 6) + (7y22 -9) -9)

= (3y= (3y22 + 7y + 7y22) + 5y + [-6 + (-9)]) + 5y + [-6 + (-9)]

= (3 + 7)y= (3 + 7)y22 + 5y + (-15) + 5y + (-15)

= 10y= 10y22 + 5y - 15 + 5y - 15

Example 2: Find (3yExample 2: Find (3y22 + 5y – 6) + 5y – 6) + (7y+ (7y22 -9) Method 2: Column -9) Method 2: Column formform

3y3y22 5y5y – – 66

++ 7y7y22 – – 99

10y10y22 5y5y – – 1515

Recall that you can subtract a rational number by adding its additive inverse or opposite. Similarly, you can subtract a polynomial by adding its additive inverse.

To find the additive inverse of a To find the additive inverse of a polynomial, replace each term with polynomial, replace each term with

its additive inverse.its additive inverse.

PolynomialPolynomial Additive InverseAdditive Inverse

x + 2yx + 2y -x – 2y-x – 2y

2x2x22 – 3x +5 – 3x +5 - 2x- 2x22 + 3x -5 + 3x -5

- 8x + 5y – 7z- 8x + 5y – 7z 8x - 5y + 7z8x - 5y + 7z

3x3x3 3 - 2x- 2x22 – 5x – 5x - 3x- 3x3 3 + 2x+ 2x22 + 5x + 5x

The additive inverse of every term must be found!!!

Example 2: Find (4xExample 2: Find (4x22 – 3y – 3y22 + + 5xy) – (8xy+ 6x5xy) – (8xy+ 6x22 + 3y + 3y22))

Method 1: Group the like terms Method 1: Group the like terms together.together. (4x(4x22 – 3y – 3y22 + 5xy) – (8xy+ 6x + 5xy) – (8xy+ 6x22 + 3y + 3y22))

= (4x= (4x22 – 3y – 3y22 + 5xy) + (– 8xy - 6x + 5xy) + (– 8xy - 6x22 - 3y - 3y22))

= (4x= (4x22 - 6x - 6x22) + (5xy – 8xy) + (- 3y) + (5xy – 8xy) + (- 3y2 2 - 3y- 3y22))

= (4 - 6)x= (4 - 6)x22 + (5 – 8)xy + (-3 - 3)y + (5 – 8)xy + (-3 - 3)y22

= -2x= -2x22 – 3xy + -6y – 3xy + -6y22

OR WOULD YOU PREFER COLUMN FORMAT?OR WOULD YOU PREFER COLUMN FORMAT?

Example 2: Find (4xExample 2: Find (4x22 – 3y – 3y22 + + 5xy) – (8xy+ 6x5xy) – (8xy+ 6x22 + 3y + 3y22) Column ) Column formatformat

4x4x22 5xy5xy -3y-3y22

-- 6x6x22 8xy8xy 3y3y22

First, reorder the terms so that the powers of x are in descending order:

(4x(4x22 + 5xy – 3y + 5xy – 3y22) – (6x) – (6x22 + 8xy+ 3y + 8xy+ 3y22))

THEN use the additive inverse to change the signsTHEN use the additive inverse to change the signs

Example 2: Find (4xExample 2: Find (4x22 – 3y – 3y22 + + 5xy) – (8xy+ 6x5xy) – (8xy+ 6x22 + 3y + 3y22) Column ) Column formatformat

4x4x22 5xy5xy -3y-3y22

++ - 6x- 6x22 - 8xy- 8xy - 3y- 3y22

- 2x- 2x22 - 3xy- 3xy - 6y- 6y22

To check this result, add -2x-2x22 – 3xy + -6y – 3xy + -6y22 and and 6x6x22 + 8xy+ + 8xy+ 3y3y22

(4x(4x22 + 5xy – 3y + 5xy – 3y22))

This is what you should get after you check it. This is what you should get after you check it.

Example 3: Find the measure of the Example 3: Find the measure of the third side of the triangle. P is the third side of the triangle. P is the measure of the perimeter.measure of the perimeter.

The perimeter is The perimeter is the sum of the the sum of the measures of the measures of the three sides of the three sides of the triangle. Let s triangle. Let s represent the represent the measure of the measure of the third side.third side.

8x2 – 8x + 5

3x 2 + 2x - 1sP = 12x2 – 7x + 9

(12x2 – 7x + 9) = (3x2 + 2x - 1) + (8x2 – 8x + 5) + s

(12x2 – 7x + 9) - (3x2 + 2x - 1) - (8x2 – 8x + 5) = s

12x2 – 7x + 9 - 3x2 - 2x + 1 - 8x2 + 8x - 5) = s

(12x2 - 3x2 - 8x2)+(– 7x - 2x + 8x) + (9 + 1 - 5) = s

x2 - x + 5 = s

The measure of the third side is x2 - x + 5.