Simplifying Polynomials 14-2 Simplifying Polynomials Course 3 Warm Up Problem of the Day Lesson Presentation

Warm Up Identify the coefficient of each monomial. 1. 3x4 2. ab 3. 4. –cb3 Use the Distributive Property to simplify each expression. 5. 9(6 + 7) 6. 4(10 – 2) 3 1 x 2 1 2 –1 117 32

Problem of the Day Warren drank 3.5 gallons of water in one week. Find the average number of ounces of water Warren drank each day that week. 64 oz

Learn to simplify polynomials.

You can simplify a polynomial by adding or subtracting like terms You can simplify a polynomial by adding or subtracting like terms. Remember that like terms have the same variables raised to the same powers. Like terms The variables have the same powers. 4a3b2 + 3a2b3 – 2a3b2 Not like terms The variables have different powers.

Additional Example 1: Identifying Like Terms Identify the like terms in each polynomial. A. 5x3 + y2 + 2 – 6y2 + 4x3 B. 3a3b2 + 3a2b3 + 2a3b2 - a3b2 5x + y + 2 – 6y + 4x 3 2 Identify like terms. Like terms: 5x3 and 4x3, y2 and –6y2 Identify like terms. 3a b + 3a b + 2a b – a b 3 2 Like terms: 3a3b2, 2a3b2, and –a3b2

Additional Example 1: Identifying Like Terms Identify the like terms in the polynomial. C. 7p3q2 + 7p2q3 + 7pq2 7p3q2 + 7p2q3 + 7pq2 Identify like terms. There are no like terms.

Identify the like terms in each polynomial. Check It Out: Example 1 Identify the like terms in each polynomial. A. 4y4 + y2 + 2 – 8y2 + 2y4 B. 7n4r2 + 3a2b3 + 5n4r2 + n4r2 4y + y + 2 – 8y + 2y 4 2 Identify like terms. Like terms: 4y4 and 2y4, y2 and –8y2 7n4r2 + 3a2b3 + 5n4r2 + n4r2 Identify like terms. Like terms: 7n4r2, 5n4r2, and n4r2

Check It Out: Example 1 Identify the like terms in the polynomial. C. 9m3n2 + 7m2n3 + pq2 9m3n2 + 7m2n3 + pq2 Identify the like terms. There are no like terms.

To simplify a polynomial, combine like terms 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.

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

Additional Example 2B: Simplifying Polynomials by Combining Like Terms 3n5m4 – 6n3m + n5m4 – 8n3m Arrange in descending order. 3n5m4 + n5m4 – 6n3m – 8n3m 3n5m4 + n5m4 – 6n3m – 8n3m Identify like terms. Combine coefficients: 3 + 1 = 4 and –6 – 8 = –14. 4n5m4 – 14n3m

Check It Out: Example 2A Simplify. 2x3+ 5x3 + 6 – 4x + 9 Arrange in descending order. 2x3+ 5x3 – 4x + 6 + 9 Identify the like terms. 2x3+ 5x3 – 4x + 6 + 9 Combine coefficients: 2 + 5 = 7 and 6 + 9 = 15 7x3 – 4x + 15

Check It Out: Example 2B Simplify. 2n5p4 – 7n6p + n5p4 – 9n6p Arrange in descending order. 2n5p4 + n5p4 – 7n6p – 9n6p Identify like terms. 2n5p4 + n5p4 – 7n6p – 9n6p Combine coefficients: 2 + 1 = 3 and –7 + –9 = –16 3n5p4 – 16n6p

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

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

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

Check It Out: Example 3A Simplify. 2(x3 + 5x2) 2(x3+ 5x2) Distributive Property 2  x3 + 2  5x2 2x3 + 10x2

Check It Out: Example 3B Simplify. –2(6m3p + 8m2p) + m2p –2(6m3p + 8m2p) + m2p Distributive Property –2  6m3p – 2  8m2p + m2p –12m3p – 16m2p + m2p –12m3p – 15m2p Combine like terms.

Additional Example 4: Business Application Pre-Algebra 14-2 Simplifying Polynomials Additional Example 4: Business Application 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. 2(r2 + rh) = 2r2 + 2 rh

Simplifying Polynomials Pre-Algebra 14-2 Simplifying Polynomials Check It Out: Example 4 Use the Distributive Property to write an equivalent expression for 3a(b2+ c). 3a(b + c) = 2 3ab + 3ac 2

Simplifying Polynomials Insert Lesson Title Here Pre-Algebra 14-2 Simplifying Polynomials Insert Lesson Title Here Lesson Quiz 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 2x and 5x , z and –3z 2 2ab2 and –5ab2, 4a2b and a2b 15x2 + 10 6k2 + 8k + 8 12mn2 + 9n