### 10-6 Dividing Polynomials

Preview

Warm Up

California Standards

Lesson Presentation

### 10-6 Dividing Polynomials

Warm Up Divide.

1. m 2

n

÷

mn

4 3. (3a + 6a 2 ) ÷ 3a 2

b

Factor each expression.

4. 5x 2 + 16x + 12 5. 16p 2 – 72p + 81 2. 2x 3

y

2 ÷ 6xy

### 10-6 Dividing Polynomials California Standards

10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, by using these techniques.

12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

### 10-6 Dividing Polynomials

To divide a polynomial by a monomial, you can first write the division as a rational expression. Then divide each term in the polynomial by the monomial.

### 10-6 Dividing Polynomials

Additional Example 1: Dividing a Polynomial by a Monomial Divide (5x 3 – 20x 2 + 30x) ÷ 5x

Write as a rational expression.

Divide each term in the polynomial by the monomial 5x.

Divide out common factors.

x

2 – 4x + 6

Simplify.

### 10-6 Dividing Polynomials

Divide.

Check It Out!

Example 1a (8p 3 – 4p 2 + 12p) ÷ ( – 4p 2 )

Write as a rational expression.

Divide each term in the polynomial by the monomial –4p 2 .

Divide out common factors.

Simplify.

### 10-6 Dividing Polynomials

Check It Out!

Example 1b Divide.

(6x 3 + 2x – 15) ÷ 6x

Write as a rational expression.

Divide each term in the polynomial by the monomial 6x.

Divide out common factors in each term.

Simplify.

### 10-6 Dividing Polynomials

Division of a polynomial by a binomial is similar to division of whole numbers.

### 10-6 Dividing Polynomials

Additional Example 2A: Dividing a Polynomial by a Binomial Divide.

x + 5

Factor the numerator.

Divide out common factors.

Simplify.

### 10-6 Dividing Polynomials

Additional Example 2B: Dividing a Polynomial by a Binomial Divide.

Factor both the numerator and denominator.

Divide out common factors.

Simplify.

### 10-6 Dividing Polynomials

Put each term of the numerator over the denominator only when the denominator is a monomial. If the denominator is a polynomial, try to factor first.

### 10-6 Dividing Polynomials

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Example 2a Divide.

k + 5

Factor the numerator.

Divide out common factors.

Simplify.

### 10-6 Dividing Polynomials

Check It Out!

Example 2b Divide.

b

– 7

Factor the numerator.

Divide out common factors.

Simplify.

### 10-6 Dividing Polynomials

Check It Out!

Example 2c Divide.

s + 6

Factor the numerator.

Divide out common factors.

Simplify.

### 10-6 Dividing Polynomials

Recall how you used long division to divide whole numbers as shown at right. You can also use long division to divide polynomials. An example is shown below.

(x 2 + 3x + 2) ÷ (x + 2) Divisor x + 2 )

x

2

x

2 x + 1 + 3x + 2 + 2x x + 2 x + 2 0 Quotient Dividend

### 10-6 Dividing Polynomials

Using Long Division to Divide a Polynomial by a Binomial

Step 1 Write the binomial and polynomial in standard form.

Step 2 Divide the first term of the dividend by the first term of the divisor. This the first term of the quotient.

Step 3 Multiply this first term of the quotient by the binomial divisor and place the product under the dividend, aligning like terms.

Step 4 Subtract the product from the dividend.

Step 5 Bring down the next term in the dividend.

Step 6 Repeat Steps 2-5 as necessary until you get 0 or until the degree of the remainder is less than the degree of the binomial.

### 10-6 Dividing Polynomials

(x 2 +10x + 21) ÷ (x + 3)

Step 1 x + 3 )

x

2 + 10x + 21

Write in long division form with expressions in standard form.

Step 2

x

+ 3 )

x

2

x

+ 10x + 21

Divide the first term of the dividend by the first term of the divisor the first term of the to get quotient.

### 10-6 Dividing Polynomials

Additional Example 3A Continued Divide using long division.

Step 3 Step 4 (x 2 +10x + 21) ÷ (x + 3)

x + 3 )

x

2

x

2

x

+ 10x + 21 + 3x

x

x + 3 )

x

2 – (

x

2 + 10x + 21 + 3x ) 0 + 7x

Multiply the first term of the quotient by the binomial divisor . Place the product under the dividend, aligning like terms. Subtract the product from the dividend.

### 10-6 Dividing Polynomials

Additional Example 3A Continued Divide using long division.

x

Step 5 x + 3 )

x

2 – (x 2 + 10x + 21 + 3x) 7x + 21

Bring down the next term in the dividend.

x + 7 Step 6 x + 3 )

x

2 – (x 2 + 10x + 21 + 3x) 7x + 21 – (7x + 21) 0

Repeat Steps 2-5 as necessary. The remainder is 0.

(x + 3)(x + 7)

x

2 + 7x + 3x + 21

x

2 + 10x + 21 

### 10-6 Dividing Polynomials

When the remainder is 0, you can check your simplified answer by multiplying it by the divisor. You should get the numerator.

### 10-6 Dividing Polynomials

Additional Example 3B: Polynomial Long Division Divide using long division.

x

– 4 )

x

2 – 2x

8

x

– 4 )

x

2 – (x 2 – x+ 2 2x

8 – 4x) – – (2x

8 8) 0

Write in long division form.

x

2 ÷ x = x

Multiply x

(x – 4). Subtract. Bring down the 8. 2x ÷ x = 2.

Multiply 2(x – 4). Subtract.

The remainder is 0.

(x + 2)(x – 4)

x

2

4x + 2x – 8

x

2

2x + 8 

### 10-6 Dividing Polynomials

Check It Out!

Example 3a Divide using long division.

(2y 2 – 5y – 3) ÷ (y – 3) Step 1 Step 2

y

– 3 ) 2y 2 – 5y – 3

Write in long division form with expressions in standard form.

y

– 3 ) 2y 2 – 2y 5y

3

Divide the first term of the dividend by the first term of the divisor the first term of the to get quotient.

### 10-6 Dividing Polynomials

Check It Out!

Example 3a Continued Divide using long division.

(2y 2 – 5y – 3) ÷ (y – 3) Step 3 Step 4

y

– 3 ) 2y 2 2y 2 – – 2y 5y 6y – 3

y

– 2y 3 ) 2y 2 – 5y

– ( 2y 2 – 6y ) 0 + y 3

Multiply the first term of the quotient by the binomial divisor . Place the product under the dividend, aligning like terms. Subtract the product from the dividend.

### 10-6 Dividing Polynomials

Check It Out!

Example 3a Continued Divide using long division.

2y

Step 5

y

– 3 ) 2y 2 – ( 2y 2 – – 5y 6y ) – 3

Bring down the next term in the dividend.

y –

3 Step 6 y – 3 )

2y

2 – (2y 2 2y + 1 – – 5y

6y)

y –

– (y

3 3 3) 0

Repeat Steps 2 –5 as necessary. The remainder is 0.

### 10-6 Dividing Polynomials

Check It Out!

Example 3a Continued Check: Multiply the answer and the divisor.

(y – 3)(2y + 1) 2y 2 + y

6y – 3 2y 2

5y – 3 

### 10-6 Dividing Polynomials

Check It Out!

Example 3b Divide using long division.

(a 2 – 8a + 12) ÷ (a – 6)

a

– 6 )

a

2 – 8a + 12

a

– 6 )

a

2 –

a

– 2 8a + 12 – (a 2 – 6a) – 2a + 12 – ( – 2a + 12) 0

Write in long division form.

a

2 ÷ a = a

Multiply a

(a – 6). Subtract. Bring down the 12. –2a ÷ a = –2.

Multiply –2(a – 6). Subtract.

The remainder is 0.

### 10-6 Dividing Polynomials

Check It Out!

Example 3b Continued Check: Multiply the answer and the divisor.

(a – 6)(a – 2)

a

2 – 2a

6a + 12

a

2

8a + 12 

### 10-6 Dividing Polynomials

Sometimes the divisor is not a factor of the dividend, so the remainder is not 0. Then the remainder can be written as a rational expression.

### 10-6 Dividing Polynomials

Additional Example 4: Long Division with a Remainder Divide (3x 2 + 19x + 26) ÷ (x + 5)

x + 5 ) 3x 2 + 19x + 26 x + 5 ) – (3x 2 3x + 4 3x 2 + 19x + 26 + 15x) 4x + 26 – (4x + 20) 6

Write in long division form.

3x 2 ÷ x = 3x.

Multiply 3x(x + 5). Subtract.

Bring down the 26. 4x ÷ x = 4.

Multiply 4(x + 5). Subtract.

The remainder is 6.

Write the remainder as a rational expression using the divisor as the denominator.

### 10-6 Dividing Polynomials

Additional Example 4 Continued Divide (3x 2 + 19x + 26) ÷ (x + 5)

Write the quotient with the remainder.

### 10-6 Dividing Polynomials

Check It Out!

Example 4a Divide.

m + 3 ) 3m 2 + 4m

2 m + 3 ) 3m – 3m 2 + 4m

– (3m 2 + 9m) 5 2 – 5m – – ( – 5m

2 15) 13

Write in long division form.

3m 2 ÷ m = 3m.

Multiply 3m(m + 3). Subtract.

Bring down the –2. –5m ÷ m = –5 .

Multiply –5(m + 3). Subtract.

The remainder is 13.

### 10-6 Dividing Polynomials

Check It Out!

Example 4a Continued Divide.

Write the remainder as a rational expression using the divisor as the denominator.

### 10-6 Dividing Polynomials

Check It Out!

Example 4b Divide.

y

– 3 )

y

2 + 3y + 2

y –

3 )

y

+ 6

y

2 + 3y + 2 – (y 2 – 3y) 6y + 2 – (6y

18) 20 y + 6 +

Write in long division form.

y 2 ÷ y = y.

Multiply y(y – 3). Subtract.

Bring down the 2. 6y ÷ y = 6.

Multiply 6(y – 3). Subtract.

The remainder is 20.

Write the quotient with the remainder.

### 10-6 Dividing Polynomials

Sometimes you need to write a placeholder for a term using a zero coefficient. This is best seen if you write the polynomials in standard form.

### 10-6 Dividing Polynomials

Additional Example 5: Dividing Polynomials That Have a Zero Coefficient Divide (x 3 – 7 – 4x) ÷ (x – 3).

(x 3 – 4x – 7) ÷ (x – 3)

x

– 3 )

x

3 + 0x 2 – 4x

7

x

– 3 )

x

3

(x 3

x

2 + 0x 2

3x 2 ) – 4x

7 3x 2 – 4x

Write the polynomials in standard form.

Write in long division form. Use 0x 2 as a placeholder for the x 2 term. x 3 ÷ x = x 2 Multiply x 2 (x – 3). Subtract.

Bring down –4x.

### 10-6 Dividing Polynomials

x

– 3 )

x

3

(x 3

– x

2 + 0x 2 3x 2 ) + 3x – + 5 4x

7 3x 2

(3x 2 –

4x 9x) 5x – (5x – – 7 15) 8

3x 3 ÷ x = 3x Multiply x 2 (x – 3). Subtract.

Bring down – 4x.

Multiply 3x(x – 3). Subtract.

Bring down – 7.

Multiply 5(x – 3). Subtract.

The remainder is 8.

(x 3 – 4x

7) ÷ (x – 3) =

### 10-6 Dividing Polynomials

Remember!

Recall from Chapter 7 that a polynomial in one variable is written in standard form when the degrees of the terms go from greatest to least.

### 10-6 Dividing Polynomials

Check It Out!

Example 5a

x

Divide (1 – 4x 2 + x 3 ) ÷ (x – 2).

(x 3 – 4x 2 + 1) ÷ (x – 2)

x

– 2 )

x

3 – 4x 2 + 0x + 1 2 )

x

3 – (x 3 – – –

x

2 4x 2 – 2x – 4 2x 2 ) + 0x + 1 2x – ( – 2x 2 2 + 0x + 4x) – 4x + 1 – ( – 4x + 8) – 7

Write in standard form.

Write in long division form. Use 0x as a placeholder for x 3 the x term.

÷ x = x 2 Multiply x 2 (x – 2). Subtract.

Bring down 0x. – 2x 2 ÷ x = –2x. Multiply –2x(x – 2). Subtract.

Bring down 1.

Multiply –4(x – 2). Subtract.

### 10-6 Dividing Polynomials

Check It Out!

Example 5a Continued Divide (1 – 4x 2 + x 3 ) ÷ (x – 2).

(1 – 4x 2 + x 3 ) ÷ (x – 2) =

### 10-6 Dividing Polynomials

Check It Out!

Example 5b Divide (4p – 1 + 2p 3 ) ÷ (p + 1).

(2p 3 + 4p – 1) ÷ (p + 1)

Write in standard form.

p + 1 ) 2p 3 + 0p

2

+ 4p – 1 p + 1 ) 2p 3 – (2p 3 2p + 0p 2 2 – 2p + 2p 2 ) + 4p + 6 – 1 – 2p 2 – ( – 2p 2 + 4p – 2p) 6p – 1 – (6p + 6) – 7

Write in long division form. Use 0p 2 as a placeholder p 3 for the p 2 ÷ p = p 2 term.

Multiply 2p 2 (p + 1). Subtract.

Bring down 4p. –2p 2 ÷ p = –2p. Multiply –2p(p + 1). Subtract.

Bring down –1.

Multiply 6(p + 1). Subtract.

### 10-6 Dividing Polynomials

Check It Out!

Example 5b Continued

(2p 3 + 4p – 1) ÷ (p + 1) =

### 10-6 Dividing Polynomials

1.

(12x 2 – 4x 2 + 20x) ÷ 4x 3x 2 – x + 5

2.

2x + 3

3.

4.

x

– 2 x + 3

### 10-6 Dividing Polynomials

Lesson Quiz: Part II Divide using long division.

5.

(x 2 + 4x + 7)  (x + 1) 6. (8x 2 + 2x 3 + 7)  (x + 3)