Warm-up

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

1. 8a 2 + 5ab 2. 3x 2 3. 5x 2 + 4x – 7/x + 7x + 2 4. 6a 2 b 2 + 7ab 5 – 6b 3 5. w 2 x 7w 3 + 6x

Warm-up

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

1. 8a 2 + 5ab binomial; 2 2. 3x 2 3. 5x 4. 6a 2 2 b + 4x – 7/x + 7x + 2 2 + 7ab 5 – 6b 3 not a polynomial not the product of # and variable trinomial; 2 trinomial; 2 5. w 2 x 7w 3 + 6x trinomial; 2

Homework 6.5

2 3 4 1 4 4 4 8 1.

2.

3.

4.

5.

6.

7.

8.

9. 4 10. 5 11. 5 + 2x 2 + 3x 3 12. 1 + 3x + 2x 2 13. - 6 + 5x + 3x 2 + x 4 14. 2 + x + 9x 2 15. - 3 + 4x – x 2 + x 3 + 3x 3 16. – 2x + x 2 - x 3 + x 4

Homework 6.5

17. 6 + 12x + 6x 2 + x 3 18. 21r 2 19. 5x 3 + 7r - 3x 2 5 x – r 2 x 2 + x + 4 – 15x 3 20. - x 3 21. 3x 3 + x 2 + x 2 - x + 1 - x + 27 22. 3x 3 23. x 3 + x 2 + x - 17 + x - 1 24. 3x 3 + x 2 - x + 64

Homework 6.5

25. - x 3 + x + 25 26. ⅓p x 3 + p 3 x 2 + px + 5p

6.6 Adding and Subtracting Polynomials

CORD Math Mrs. Spitz Fall 2006

Objectives:

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

Assignment:

 6.6 Worksheet

Application:

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

x – 3 in.

Application:

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

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

Application

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

Example 1: Find (3y 2 (7y 2 -9) + 5y – 6) +

Method 1: Group the like terms together.

(3y 2 + 5y – 6) + (7y 2 = (3y 2 + 7y 2 -9) ) + 5y + [-6 + (-9)] = (3 + 7)y 2 = 10y 2 + 5y + (-15) + 5y - 15

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

+ 3y 2 7y 2 10y 2 5y 5y – 6 – 9 – 15 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 polynomial, replace each term with its additive inverse.

Polynomial

x + 2y 2x 2 – 3x +5 - 8x + 5y – 7z 3x 3 - 2x 2 – 5x

Additive Inverse

-x – 2y - 2x 2 + 3x -5 8x - 5y + 7z - 3x 3 + 2x 2 + 5x

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

Example 2: Find (4x 2 – (8xy+ 6x 2 + 3y 2 ) – 3y 2 + 5xy)

Method 1: Group the like terms together.

(4x 2 – 3y 2 = (4x 2 – 3y + 5xy) – (8xy+ 6x 2 2 + 3y + 5xy) + (– 8xy - 6x 2 2 ) - 3y 2 ) = (4x 2 - 6x 2 ) + (5xy – 8xy) + (- 3y 2 - 3y 2 ) = (4 - 6)x 2 + (5 – 8)xy + (-3 - 3)y 2 = -2x 2 – 3xy + -6y 2 OR WOULD YOU PREFER COLUMN FORMAT?

Example 2: Find (4x – (8xy+ 6x 2 2 – 3y 2 + 5xy) + 3y 2 ) Column format

4x 2 6x 2 5xy 8xy -3y 2 3y 2 First, reorder the terms so that the powers of x are in descending order:

(4x 2 + 5xy – 3y 2 ) – (6x 2 + 8xy+ 3y 2 ) THEN use the additive inverse to change the signs

Example 2: Find (4x – (8xy+ 6x 2 2 – 3y 2 + 5xy) + 3y 2 ) Column format

+ 4x 2 - 6x 2 - 2x 2 5xy - 8xy - 3xy -3y 2 - 3y 2 - 6y 2 To check this result, add -2x 2 – 3xy + -6y 2 and

6x 2 + 8xy+ 3y 2 (4x 2 + 5xy – 3y 2 ) This is what you should get after you check it.

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

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

8x 2 – 8x + 5 P = 12x 2 – 7x + 9

(12x 2 – 7x + 9) = (3x 2 + 2x - 1) + (8x 2 – 8x + 5) + s (12x 2 – 7x + 9) - (3x 2 + 2x - 1) - (8x 2 – 8x + 5) = s 12x 2 – 7x + 9 - 3x 2 - 2x + 1 - 8x 2 + 8x - 5) = s (12x 2 - 3x 2 - 8x 2 )+(– 7x - 2x + 8x) + (9 + 1 - 5) = s x 2 - x + 5 = s The measure of the third side is x 2 - x + 5.