1. Simplifying Polynomials 14-2 Warm Up Identify the coefficient of each monomial. 1. 3x 4 2. ab 3. 4. –cb 3 Use the Distributive Property to simplify each expression. 5. 9(6 + 7)6. 4(10 – 2) 3 1 –1 x 2 1 2 117 32

2. Simplifying Polynomials 14-2 Learn to simplify polynomials.

3. Simplifying Polynomials 14-2 You can simplify a polynomial by adding or subtracting like terms. Remember that like terms have the same variables raised to the same powers. 4a 3 b 2 + 3a 2 b 3 – 2a 3 b 2 Like terms Not like terms The variables have the same powers. The variables have different powers.

4. Simplifying Polynomials 14-2 Example 1: Identifying Like Terms Identify the like terms in each polynomial. A. 5x 3 + y 2 + 2 – 6y 2 + 4x 3 B. 3a 3 b 2 + 3a 2 b 3 + 2a 3 b 2 - a 3 b 2 Identify like terms. 5x + y + 2 – 6y + 4x 3 2 2 3 Like terms: 5x 3 and 4x 3, y 2 and –6y 2 3a b + 3a b + 2a b – a b 3 2 2 3 3 3 2 2 Identify like terms. Like terms: 3a 3 b 2, 2a 3 b 2, and –a 3 b 2

5. Simplifying Polynomials 14-2 Example 1: Identifying Like Terms Identify the like terms in the polynomial. C. 7p 3 q 2 + 7p 2 q 3 + 7pq 2 Identify like terms. There are no like terms. 7p 3 q 2 + 7p 2 q 3 + 7pq 2

6. Simplifying Polynomials 14-2 Example 2 Identify the like terms in each polynomial. A. 4y 4 + y 2 + 2 – 8y 2 + 2y 4 B. 7n 4 r 2 + 3a 2 b 3 + 5n 4 r 2 + n 4 r 2 Identify like terms. 4y + y + 2 – 8y + 2y 4 2 2 4 Like terms: 4y 4 and 2y 4, y 2 and –8y 2 7n 4 r 2 + 3a 2 b 3 + 5n 4 r 2 + n 4 r 2 Identify like terms. Like terms: 7n 4 r 2, 5n 4 r 2, and n 4 r 2

7. Simplifying Polynomials 14-2 Example 2 Identify the like terms in the polynomial. C. 9m 3 n 2 + 7m 2 n 3 + pq 2 Identify the like terms. There are no like terms. 9m 3 n 2 + 7m 2 n 3 + pq 2

8. Simplifying Polynomials 14-2 To simplify a polynomial, combine like terms. It may be easier to arrange the terms in descending order (highest degree to lowest degree) before combining like terms.

9. Simplifying Polynomials 14-2 Example 3A: Simplifying Polynomials by Combining Like Terms Simplify. 4x 2 + 2x 2 + 7 – 6x + 9 Identify like terms. 4x 2 + 2x 2 – 6x + 7 + 9 2 6x – 6x + 16 Combine coefficients: 4 + 2 = 6 and 7 + 9 = 16 Arrange in descending order. 4x 2 + 2x 2 – 6x + 7 + 9

10. Simplifying Polynomials 14-2 Example 3B: Simplifying Polynomials by Combining Like Terms Simplify. 3n 5 m 4 – 6n 3 m + n 5 m 4 – 8n 3 m Arrange in descending order. Identify like terms. 3n 5 m 4 + n 5 m 4 – 6n 3 m – 8n 3 m Combine coefficients: 3 + 1 = 4 and –6 – 8 = –14. 3n 5 m 4 + n 5 m 4 – 6n 3 m – 8n 3 m 4n 5 m 4 – 14n 3 m

11. Simplifying Polynomials 14-2 Example 4A Simplify. 2x 3 + 5x 3 + 6 – 4x + 9 Identify the like terms.Combine coefficients: 2 + 5 = 7 and 6 + 9 = 15 Arrange in descending order. 2x 3 + 5x 3 – 4x + 6 + 9 7x 3 – 4x + 15

12. Simplifying Polynomials 14-2 Example 4B Simplify. 2n 5 p 4 – 7n 6 p + n 5 p 4 – 9n 6 p Arrange in descending order. Identify like terms. Combine coefficients: 2 + 1 = 3 and –7 + –9 = –16 2n 5 p 4 + n 5 p 4 – 7n 6 p – 9n 6 p 3n 5 p 4 – 16n 6 p

13. Simplifying Polynomials 14-2 Sometimes you may need to use the Distributive Property to simplify a polynomial.

14. Simplifying Polynomials 14-2 Example 5A: Simplifying Polynomials by Using the Distributive Property Simplify. 3(x 3 + 5x 2 ) 2 3(x + 5x ) 3 Distributive Property3  x 3 + 3  5x 2 2 3x + 15x 3

15. Simplifying Polynomials 14-2 Example 5B: Simplifying Polynomials by Using the Distributive Property Simplify. –4(3m 3 n + 7m 2 n) + m 2 n Distributive Property –4(3m 3 n + 7m 2 n) + m 2 n –4  3m 3 n – 4  7m 2 n + m 2 n –12m 3 n – 28m 2 n + m 2 n –12m 3 n – 27m 2 n Combine like terms.

16. Simplifying Polynomials 14-2 Example 6A Simplify. 2(x 3 + 5x 2 ) Distributive Property 2  x 3 + 2  5x 2 2x 3 + 10x 2

17. Simplifying Polynomials 14-2 Example 6B Simplify. –2(6m 3 p + 8m 2 p) + m 2 p Distributive Property –2(6m 3 p + 8m 2 p) + m 2 p –2  6m 3 p – 2  8m 2 p + m 2 p –12m 3 p – 16m 2 p + m 2 p –12m 3 p – 15m 2 p Combine like terms.

18. Simplifying Polynomials 14-2 Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

19. Simplifying Polynomials 14-2 Lesson Quiz Identify the like terms in each polynomial. 1. 2x 2 – 3z + 5x 2 + z + 8z 2 2. 2ab 2 + 4a 2 b – 5ab 2 – 4 + a 2 b Simplify. 3. 5(3x 2 + 2) 4. –2k 2 + 10 + 8k 2 + 8k – 2 5. 3(2mn 2 + 3n) + 6mn 2 2x and 5x, z and –3z 22 2ab 2 and –5ab 2, 4a 2 b and a 2 b 15x 2 + 10 6k 2 + 8k + 8 12mn 2 + 9n

20. Simplifying Polynomials 14-2 1. Identify the like terms in the given polynomial. 3x 3 – 7y +6x 3 + y + 9y 2 A. 3x 3 and 6x 3, –7y and 9y 2 B. 3x 3 and 9y 2, y and –7y C. 3x 3 and 9y 2, –7y and 9y 2 D. 3x 3 and 6x 3, y and –7y Lesson Quiz for Student Response Systems

21. Simplifying Polynomials 14-2 2. Identify the like terms in the given polynomial. 4pq 2 + 6p 2 q – 7pq 2 – 11 + p 2 q A. 4pq 2 and –7pq 2, 4pq 2 and p 2 q B. 4pq 2 and –7pq 2, 6p 2 q and p 2 q C. 6p 2 q and –7pq 2, 4pq 2 and p 2 q D. 6p 2 q and –7pq 2, 6p 2 q and p 2 q Lesson Quiz for Student Response Systems

22. Simplifying Polynomials 14-2 3. Simplify. 7(4p 2 + 3) A. 11p 2 + 21 B. 11p 2 + 10 C. 28p 2 + 10 D. 28p 2 + 21 Lesson Quiz for Student Response Systems

23. Simplifying Polynomials 14-2 4. Simplify. –5t 2 + 7 + 11t 2 + 9t – 4 A. 16t 2 + 9t – 3 B. 6t 2 + 16t – 4 C. 6t 2 + 9t + 3 D. 16t 2 + 9t + 11 Lesson Quiz for Student Response Systems