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Simplifying Polynomials

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  1. Simplifying Polynomials 13-2

  2. Warm Up x 1 2 2 Identify the coefficient of each monomial. 1.3x42.ab 3.4. –cb3 Use the Distributive Property to simplify each expression. 5. 9(6 + 7) 6. 4(10 – 2) 3 1 –1 32 117

  3. Learn to simplify polynomials.

  4. Examples: Identifying Like Terms 3 2 2 3 5x + y + 2 – 6y + 4x Like terms: 5x3 and 4x3, y2 and –6y2 Like terms: 3a3b2, 2a3b2, and –a3b2 3 2 3 2 2 3 2 3 3a b + 3a b + 2a b – a b Identify like terms. Identify like terms. Identify the like terms in each polynomial. A. 5x3 + y2 + 2 – 6y2 + 4x3 B. 3a3b2 + 3a2b3 + 2a3b2 - a3b2

  5. Example: Identifying Like Terms 7p3q2 + 7p2q3 + 7pq2 There are no like terms. Identify like terms. Identify the like terms in the polynomial. C. 7p3q2 + 7p2q3 + 7pq2

  6. Try This 4 2 2 4 4y + y + 2 – 8y + 2y Like terms: 4y4 and 2y4, y2 and –8y2 7n4r2 + 3a2b3 + 5n4r2 + n4r2 Identify like terms. Identify like terms. Identify the like terms in each polynomial. A. 4y4 + y2 + 2 – 8y2 + 2y4 B. 7n4r2 + 3a2b3 + 5n4r2 + n4r2 Like terms: 7n4r2, 5n4r2, and n4r2

  7. Try This There are no like terms. Identify the like terms. Identify the like terms in the polynomial. C. 9m3n2 + 7m2n3 + pq2 9m3n2 + 7m2n3 + pq2

  8. 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. Example: Simplifying Polynomials by Combining Like Terms 4x2 + 2x2– 6x + 7 + 9 2 6x – 6x + 16 Identify like terms. Combine coefficients: 4 + 2 = 6 and 7 + 9 = 16 Arrange in descending order. Simplify. A. 4x2 + 2x2 + 7 – 6x + 9 4x2 + 2x2 – 6x + 7 + 9

  10. Example: Simplifying Polynomials by Combining Like Terms 3n5m4+ n5m4 –6n3m – 8n3m 3n5m4+ n5m4 –6n3m – 8n3m 4n5m4–14n3m Arrange in descending order. Identify like terms. Combine coefficients: 3 + 1 = 4 and –6 – 8 = – 14. Simplify. B. 3n5m4–6n3m + n5m4 – 8n3m

  11. Try This Identify the like terms. Combine coefficients: 2 + 5 = 7 and 6 + 9 = 15 Arrange in descending order. Simplify. A. 2x3+ 5x3 + 6 – 4x + 9 2x3+ 5x3 – 4x + 6 + 9 2x3+ 5x3 – 4x + 6 + 9 7x3– 4x + 15

  12. Try This Arrange in descending order. Identify like terms. Combine coefficients: 2 + 1 = 3 and –7 + –9 = –16 Simplify. B. 2n5p4–7n6p + n5p4 – 9n6p 2n5p4+ n5p4 –7n6p – 9n6p 2n5p4+ n5p4 –7n6p – 9n6p 3n5p4–16n6p

  13. Sometimes you may need to use the Distributive Property to simplify a polynomial.

  14. Example: Simplifying Polynomials by Using the Distributive Property 2 3 3(x + 5x ) 2 3 3 x + 3  5x 2 3 3x + 15x Distributive Property Simplify. A. 3(x3 + 5x2)

  15. Example: Simplifying Polynomials by Using the Distributive Property –4(3m3n + 7m2n) + m2n –4  3m3n – 4  7m2n + m2n –12m3n – 28m2n + m2n –12m3n – 27m2n Distributive Property Combine like terms. Simplify. B. –4(3m3n + 7m2n) + m2n

  16. Try This 2(x3+ 5x2) 2  x3 + 2  5x2 2x3 + 10x2 Distributive Property Simplify. A. 2(x3 + 5x2)

  17. Try This –2(6m3p + 8m2p) + m2p –2  6m3p – 2  8m2p + m2p –12m3p – 16m2p + m2p –12m3p – 15m2p Distributive Property Combine like terms. Simplify. B. –2(6m3p + 8m2p) + m2p

  18. Example: Business Application 2 2 2(r + rh) = 2r + 2rh The surface area of a right cylinder can be found by using the expression 2(r2 + rh), where r is the radius and h is the height. Use the Distributive Property to write an equivalent expression.

  19. Try This 2 2 3a(b + c) = 3ab + 3ac Use the Distributive Property to write an equivalent expression for 3a(b2+ c).

  20. Lesson Quiz 2 2 2x and 5x , z and –3z 2ab2 and –5ab2, 4a2b and a2b 15x2 + 10 6k2 + 8k + 8 12mn2 + 9n 3h3–5h2 + 8h – 9 Identify the like terms in each polynomial. 1. 2x2 – 3z + 5x2 + z + 8z2 2. 2ab2 + 4a2b – 5ab2 – 4 + a2b Simplify. 3. 5(3x2 + 2) 4. –2k2 + 10 + 8k2 + 8k – 2 5. 3(2mn2 + 3n) + 6mn2 6. 4h2 + 3h3 – 7 – 9h2 + 8h – 2