#### Transcript Chapter 10 Section 6

**10-6 Dividing Polynomials**

**Preview**

**10-6 Dividing Polynomials**

**Warm Up Divide.**

**1. ***m* 2

*n *

÷

*mn*

4 **3. **(3*a *+ 6*a* 2 ) ÷ 3*a* 2

*b*

**Factor each expression.**

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

*y*

2 ÷ 6*xy*

**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 (5 x 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 – 4*x *+ 6

*Simplify.*

**10-6 Dividing Polynomials**

**Divide.**

**Check It Out!**

**Example 1a (8 p 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.**

**(6 x 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**

**Helpful Hint**

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**

**Check It Out!**

**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**

**Additional Example 3A: Polynomial Long Division Divide using long division. Check your answer.**

**( x 2 +10x + 21) ÷ (x + 3)**

**Step 1 ***x *+ 3 )

*x*

2 + 10*x + *21

*Write in long division form with expressions in standard form.*

**Step 2**

*x*

+ 3 )

*x*

2

*x*

+ 10*x + *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*

+ 10*x + *21 + 3*x*

*x*

*x *+ 3 )

*x*

2 – (

*x*

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

*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 + 10*x * *+ *21 + 3*x*) 7*x* *+ *21

*Bring down the next term in the dividend.*

*x + *7 **Step 6 ***x *+ 3 )

*x*

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

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

**10-6 Dividing Polynomials**

**Additional Example 3A Continued Check: Multiply the answer and the divisor.**

(*x *+ 3)(*x *+ 7)

*x*

2 *+ *7*x *+ 3*x *+ 21

*x*

2 *+ *10*x *+ 21

**10-6 Dividing Polynomials**

**Helpful Hint**

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 – 2*x *

*–*

8

*x*

– 4 )

*x*

2 – (*x* 2 – *x*+ 2 2*x *

*–*

8 – 4*x*) – – (2*x *

*–*

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.*

**10-6 Dividing Polynomials**

**Additional Example 3B Continued Check: Multiply the answer and the divisor.**

(*x *+ 2)(*x* – 4)

*x*

2

*–*

4*x *+ 2*x * – 8

*x*

2

*–*

2*x *+ 8

**10-6 Dividing Polynomials**

**Check It Out!**

**Example 3a Divide using long division.**

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

*y*

– 3 ) 2*y* 2 – 5*y* – 3

*Write in long division form with expressions in standard form.*

*y*

– 3 ) 2*y* 2 – 2*y* 5*y *

*–*

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.**

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

*y*

– 3 ) 2*y* 2 2*y* 2 – – 2*y* 5*y* 6*y* – 3

*y*

– 2*y* 3 ) 2*y* 2 – 5*y *

*–*

– ( 2*y* 2 – 6*y* ) 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.**

2*y*

**Step 5**

*y*

– 3 ) 2*y* 2 – ( 2*y* 2 – – 5*y* 6*y* ) – 3

*Bring down the next term in the dividend.*

*y –*

3 **Step 6 ***y* – 3 )

*2y*

2 – (2*y* 2 2*y * + 1 – – 5*y *

*–*

6*y*)

*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)(2*y *+ 1) 2*y* 2 + *y *

*–*

6*y* – 3 2*y* 2

*–*

5*y* – 3

**10-6 Dividing Polynomials**

**Check It Out!**

**Example 3b Divide using long division.**

**( a 2 – 8a + 12) ÷ (a – 6)**

*a*

– 6 )

*a*

2 – 8*a + *12

*a*

– 6 )

*a*

2 –

*a *

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

*–*

6*a *+ 12

*a*

2

*–*

8*a *+ 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 (3 x 2 + 19x + 26) ÷ (x + 5)**

*x *+ 5 ) 3*x* 2 + 19*x + *26 *x *+ 5 ) – (3*x* 2 3*x* + 4 3*x* 2 + 19*x + *26 + 15*x*) 4*x* + 26 – (4*x + *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 (3 x 2 + 19x + 26) ÷ (x + 5)**

*Write the quotient with the remainder.*

**10-6 Dividing Polynomials**

**Check It Out!**

**Example 4a Divide.**

*m *+ 3 ) 3*m* 2 + 4*m *

*–*

2 *m *+ 3 ) 3*m* – 3*m* 2 + 4*m *

*–*

– (3*m* 2 + 9*m*) 5 2 – 5*m* – – ( – 5*m *

*–*

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 + 3*y + *2

*y –*

3 )

*y*

+ 6

*y*

2 + 3*y + *2 – (*y* 2 – 3*y*) 6*y * + 2 – (6*y *

*–*

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 – 4*x* – 7) ÷ (*x* – 3)

*x*

– 3 )

*x*

3 + 0*x* 2 – 4*x *

*–*

7

*x*

– 3 )

*x*

3

*–*

(*x* 3

*x*

2 + 0*x* 2

*–*

3*x* 2 ) – 4*x *

*–*

7 3*x* 2 – 4*x*

*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**

**Additional Example 5 Continued**

*x*

– 3 )

*x*

3

*–*

(*x* 3

*– x*

2 + 0*x* 2 3*x* 2 ) + 3*x* – + 5 4*x *

*–*

7 3*x* 2

*–*

(3*x* 2 –

*–*

4*x* 9*x*) 5*x* – (5*x* – – 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 – 4*x *

*–*

*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 – 4 x 2 + x 3 ) ÷ (x – 2).**

(*x* 3 – 4*x* 2 + 1) ÷ (*x* – 2)

*x*

– 2 )

*x*

3 – 4x 2 + 0*x* + 1 2 )

*x*

3 – (*x* 3 – – –

*x*

2 4*x* 2 – 2*x* – 4 2*x* 2 ) + 0*x *+ 1 2*x* – ( – 2*x* 2 2 + 0*x* + 4*x*) – 4*x * + 1 – ( – 4*x *+ 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 – 4 x 2 + x 3 ) ÷ (x – 2).**

(1 – 4*x* 2 + *x* 3 ) ÷ (*x* – 2) =

**10-6 Dividing Polynomials**

**Check It Out!**

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

(2*p* 3 + 4*p* – 1) ÷ (*p *+ 1)

*Write in standard form.*

*p *+ 1 ) 2*p* 3 + 0*p*

*2*

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

(2*p* 3 + 4*p* – 1) ÷ (*p *+ 1) =

**10-6 Dividing Polynomials**

**Lesson Quiz: Part I Add or Subtract. Simplify your answer.**

**1. **

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

**2.**

2*x *+ 3

**3.**

**4.**

*x *

– 2 *x *+ 3

**10-6 Dividing Polynomials**

**Lesson Quiz: Part II Divide using long division.**

**5.**

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