2. Example 1 Add Polynomials Vertically and Horizontally a.Add and in a vertical format. 2x 32x 3 x9 + 4x 24x 2 + – x 3x 3 5x5x1 6x 26x 2 – + – b. Add and in a horizontal format. 3x 33x 3 4x4x8 2x 22x 2 – ++ SOLUTION 2x 32x 3 x9 + 4x 24x 2 + – x 3x 3 5x5x1 6x 26x 2 – + – + a. 5x 25x 2 2x2x + 4x 24x 2 3 7x7x – + b. 5x 25x 2 2x2x + 4x 24x 2 3 7x7x – + () + () 5x 25x 2 4x 24x 2 + 2x2x 3 7x7x – + () + () = 9x 29x 2 3 5x5x – + =

3. Example 2 Subtract Polynomials Vertically and Horizontally a. Subtract from in a vertical format. 3x 33x 3 9x9x 4x 24x 2 + + 6x 36x 3 7x7x12 x 2x 2 – + – SOLUTION a. Align like terms, then add the opposite of the subtracted polynomial. b. Subtract from in a horizontal format. 9x 39x 3 11 x – + 2x 32x 3 10x 5x 25x 2 –– 6x 36x 3 7x7x12 x 2x 2 – + – 3x 33x 3 9x9x 4x 24x 2 + + () – 6x 36x 3 7x7x x 2x 2 – + – 3x 33x 3 9x9x 4x 24x 2 + ––– 3x 33x 3 2x2x 5x 25x 2 –– –

4. Example 2 Subtract Polynomials Vertically and Horizontally b. Write the opposite of the subtracted polynomial, then add like terms. 9x 39x 3 11 x – + 2x 32x 3 10x 5x 25x 2 –– ()() – 9x 39x 3 11 x + 2x 32x 3 10x 5x 25x 2 ––– = – Distributive property 11 x + 2x 32x 3 10x 9x 39x 3 –– = – Group like terms. () 5x 25x 2 – )( ++ () 11 9x9x 7x 37x 3 –– = – Combine like terms. 5x 25x 2 –

5. Example 3 Use the Distributive Property Simplify the expression. 2x 22x 2 5 x – + x 2x 2 7 3x3x + – ()() + 42 14 6x6x + 8x 28x 2 20 4x4x – = – Use distributive property. ++ 2x 22x 2 6x6x + 8x 28x 2 4x4x 2x 22x 2 – = Group like terms. () + () ++ () 20 – 14 10x 2 = Combine like terms. 2x2x + 6 + a. x 3x 3 1 x + () b. xx 2x 2 ++ – x 3x 3 1 x + () x 2x 2 + –

6. Example 3 Use the Distributive Property = Use distributive property. x 4x 4 x 3x 3 ++ x 2x 2 x + x 3x 3 x 2x 2 x + –– 1 – = Group like terms. x 4x 4 x 3x 3 ++ 1 – () x 3x 3 – x 2x 2 + () x 2x 2 – x ( x ) – = Combine like terms. x 4x 4 2x2x + 1 –

7. How has the enrollment of students in public high schools changed?

10. Example 4 Find a Polynomial Model High School Enrollment The number of students (in thousands) enrolled in high schools in the United States from 1990 to 2002 can be approximated by the models shown, where t is the number of years since 1990. 3t 23t 2 11,281 258t + – + 2t 22t 2 1149 t + – Public:Private: a. Write a model for the total number of high school students E from 1990 to 2002. b. Use the model to estimate the high school enrollment in 2000.

11. Example 4 Find a Polynomial Model SOLUTION a. To write a model for the total number, add the polynomials. 3t 23t 2 11,281 258t + – + 2t 22t 2 1149 t + – () + () = E3t 23t 2 11,281 2t 22t 2 + – + 1149 + () + ) = ) 258t ( – t ( Group like terms. t 2t 2 257t + – + = 12,430 Combine like terms. ANSWER A model for the total number of students is t 2t 2 257t + – + 12,430.

12. Example 4 Find a Polynomial Model b. In this model, t 10 represents 2000. So, substitute 10 for t in the model. ANSWER The high school enrollment in 2000 was about 14.9 million. = E t 2t 2 257t + – + = 12,430 10 12,430 + – ( + = 257 Substitute 10 for t. )2)2 10 () = 14,900 Simplify.

13. Checkpoint 1. Perform the indicated operation. Write the answer in standard form. Simplify Polynomial Expressions x 2x 2 9 4x4x – ++ () ANSWER 4x 24x 2 14 x ++ ANSWER x 2x 2 12 6x6x + x 3x 3 + – 3x 23x 2 5 5x5x ++ () 2. 4x 24x 2 5 8x8x + () x 3x 3 + – 3x 23x 2 7 2x2x ++ () –

14. Checkpoint 3. Simplify the expression. Simplify Polynomial Expressions x 2x 2 3 + 3x 23x 2 1 ()() 2 –– 4 ANSWER 2x 22x 2 14 – 4. x 2x 2 x ++ () 2 + 5x 2x 2 x () 42 –– ANSWER 7x 27x 2 2 3x3x ++

22. Let the number of bald eagle nesting pairs from 1996 to 2005 in Yellowstone National Park be E and the number of peregrine falcon nesting pairs be P. These numbers can be modeled by the following equations where t is the number of years since 1996. Write a model that represents the total number of bald eagle and peregrine falcon nesting pairs from 1996 to 2005. Use the model to estimate the total number of bald eagle and peregrine falcon nesting pairs in 2002.

23. Perimeter Use the figure and the given value of the perimeter P to write an equation for P in terms of x. Then find the value of x.