Simplifying Polynomials14-2

Warm UpIdentify 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

–1x2

12

117 32

Simplifying Polynomials14-2

Learn to simplify polynomials.

Simplifying Polynomials14-2

You can simplify a polynomial by adding or subtracting like terms. Remember that like terms have the same variables raised to the same powers.

4a3b2 + 3a2b3 – 2a3b2

Like terms

Not like terms

The variables have the same powers.

The variables have different powers.

Simplifying Polynomials14-2

Example 1: Identifying Like Terms

Identify the like terms in each polynomial.

A. 5x3 + y2 + 2 – 6y2 + 4x3

B. 3a3b2 + 3a2b3 + 2a3b2 - a3b2

Identify like terms. 5x + y + 2 – 6y + 4x3 2 2 3

Like terms: 5x3 and 4x3, y2 and –6y2

3a b + 3a b + 2a b – a b3 2 2 33 3 2 2 Identify like terms.

Like terms: 3a3b2, 2a3b2, and –a3b2

Simplifying Polynomials14-2

Example 1: Identifying Like Terms

Identify the like terms in the polynomial.

C. 7p3q2 + 7p2q3 + 7pq2

Identify like terms.

There are no like terms.

7p3q2 + 7p2q3 + 7pq2

Simplifying Polynomials14-2

Example 2

Identify the like terms in each polynomial.

A. 4y4 + y2 + 2 – 8y2 + 2y4

B. 7n4r2 + 3a2b3 + 5n4r2 + n4r2

Identify like terms. 4y + y + 2 – 8y + 2y4 2 2 4

Like terms: 4y4 and 2y4, y2 and –8y2

7n4r2 + 3a2b3 + 5n4r2 + n4r2 Identify like terms.

Like terms: 7n4r2, 5n4r2, and n4r2

Simplifying Polynomials14-2

Example 2

Identify the like terms in the polynomial.

C. 9m3n2 + 7m2n3 + pq2

Identify the like terms.

There are no like terms.

9m3n2 + 7m2n3 + pq2

Simplifying Polynomials14-2

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.

Simplifying Polynomials14-2

Example 3A: Simplifying Polynomials by Combining Like Terms

Simplify.

4x2 + 2x2 + 7 – 6x + 9

Identify like terms. 4x2 + 2x2– 6x + 7 + 9

26x – 6x + 16 Combine coefficients: 4 + 2 = 6 and 7 + 9 = 16

Arrange in descending order.

4x2 + 2x2 – 6x + 7 + 9

Simplifying Polynomials14-2

Example 3B: Simplifying Polynomials by Combining Like Terms

Simplify.

3n5m4 – 6n3m + n5m4 – 8n3m

Arrange in descending order.

Identify like terms.

3n5m4 + n5m4 – 6n3m – 8n3m

Combine coefficients: 3 + 1 = 4 and –6 – 8 = –14.

3n5m4 + n5m4 – 6n3m – 8n3m

4n5m4 – 14n3m

Simplifying Polynomials14-2

Example 4A

Simplify.

2x3+ 5x3 + 6 – 4x + 9

Identify the like terms. Combine coefficients: 2 + 5 = 7 and 6 + 9 = 15

Arrange in descending order.

2x3+ 5x3 – 4x + 6 + 9

2x3+ 5x3 – 4x + 6 + 9

7x3 – 4x + 15

Simplifying Polynomials14-2

Example 4B

Simplify.

2n5p4 – 7n6p + n5p4 – 9n6p

Arrange in descending order.

Identify like terms.

Combine coefficients: 2 + 1 = 3 and –7 + –9 = –16

2n5p4 + n5p4 – 7n6p – 9n6p

2n5p4 + n5p4 – 7n6p – 9n6p

3n5p4 – 16n6p

Simplifying Polynomials14-2

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

Simplifying Polynomials14-2

Example 5A: Simplifying Polynomials by Using the Distributive Property

Simplify.

3(x3 + 5x2)

23(x + 5x )3 Distributive Property

3 x3 + 3 5x2

23x + 15x 3

Simplifying Polynomials14-2

Example 5B: Simplifying Polynomials by Using the Distributive Property

Simplify.

–4(3m3n + 7m2n) + m2n

Distributive Property –4(3m3n + 7m2n) + m2n

–4 3m3n – 4 7m2n + m2n

–12m3n – 28m2n + m2n

–12m3n – 27m2n Combine like terms.

Simplifying Polynomials14-2

Example 6A

Simplify.

2(x3 + 5x2)

2(x3+ 5x2) Distributive Property

2 x3 + 2 5x2

2x3 + 10x2

Simplifying Polynomials14-2

Example 6B

Simplify.

–2(6m3p + 8m2p) + m2p

Distributive Property –2(6m3p + 8m2p) + m2p

–2 6m3p – 2 8m2p + m2p

–12m3p – 16m2p + m2p

–12m3p – 15m2p Combine like terms.

Simplifying Polynomials14-2

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

Simplifying Polynomials14-2

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 –3z2 2

2ab2 and –5ab2, 4a2b and a2b

15x2 + 10

6k2 + 8k + 8

12mn2 + 9n

Simplifying Polynomials14-2

1. Identify the like terms in the given polynomial. 3x3 – 7y +6x3 + y + 9y2

A. 3x3 and 6x3, –7y and 9y2

B. 3x3 and 9y2, y and –7y

C. 3x3 and 9y2, –7y and 9y2

D. 3x3 and 6x3, y and –7y

Lesson Quiz for Student Response Systems

Simplifying Polynomials14-2

2. Identify the like terms in the given polynomial. 4pq2 + 6p2q – 7pq2 – 11 + p2q

A. 4pq2 and –7pq2, 4pq2 and p2q

B. 4pq2 and –7pq2, 6p2q and p2q

C. 6p2q and –7pq2, 4pq2 and p2q

D. 6p2q and –7pq2, 6p2q and p2q

Lesson Quiz for Student Response Systems

Simplifying Polynomials14-2

3. Simplify. 7(4p2 + 3)

A. 11p2 + 21

B. 11p2 + 10

C. 28p2 + 10

D. 28p2 + 21

Lesson Quiz for Student Response Systems

Simplifying Polynomials14-2

4. Simplify. –5t2 + 7 + 11t2 + 9t – 4

A. 16t2 + 9t – 3

B. 6t2 + 16t – 4

C. 6t2 + 9t + 3

D. 16t2 + 9t + 11

Lesson Quiz for Student Response Systems