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10-6 Dividing Polynomials

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Lesson Presentation

California Standards

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Warm UpDivide.

1. m2n ÷ mn4 2. 2x3y2 ÷ 6xy

3. (3a + 6a2) ÷ 3a2b

Factor each expression.

4. 5x2 + 16x + 12

5. 16p2 – 72p + 81


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.


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 PolynomialsAdditional Example 1: Dividing a Polynomial by a

Monomial

Divide (5x3 – 20x2 + 30x) ÷ 5x

x2 – 4x + 6

Write as a rational expression.

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

Divide out common factors.

Simplify.


Check It Out! Example 1a Divide.

(8p3 – 4p2 + 12p) ÷ (–4p2)


Divide each term in the polynomial by the monomial –4p2.


Simplify.


Check It Out! Example 1b

Divide.

(6x3 + 2x – 15) ÷ 6x


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

Divide out common factors in each term.

Simplify.


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


Additional Example 2A: Dividing a Polynomial by a Binomial

Divide.

x + 5

Factor the numerator.


Simplify.

10-6 Dividing PolynomialsAdditional Example 2B: Dividing a Polynomial by a

BinomialDivide.

Factor both the numerator and denominator.


Simplify.


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.

Helpful Hint


Check It Out! Example 2a

Divide.

k + 5



Simplify.

10-6 Dividing PolynomialsCheck It Out! Example 2b

Divide.

b – 7



Simplify.


Check It Out! Example 2c

Divide.

s + 6



Simplify.


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.

) x2 + 3x + 2x + 1

x2 + 2xx + 2x + 2

0

x + 2

(x2 + 3x + 2) ÷ (x + 2)

Divisor Quotient

Dividend


Using Long Division to Divide a Polynomial by a Binomial

Step 1 Write the binomial and polynomial in standard form.

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

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

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 PolynomialsAdditional Example 3A: Polynomial Long Division

Divide using long division. Check your answer.

(x2 +10x + 21) ÷ (x + 3)

x2 + 10x + 21)Step 1 x + 3Write in long division form

with expressions in standard form.

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

x2 + 10x + 21)Step 2 x + 3x

10-6 Dividing PolynomialsAdditional Example 3A Continued

Divide using long division.

(x2 +10x + 21) ÷ (x + 3)

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

x2 + 10x + 21)Step 3 x + 3x

x2 + 3x

x2 + 10x + 21)Step 4 x + 3–(x2 + 3x)

x

0 + 7x

Subtract the product from the dividend.


Additional Example 3A Continued


x2 + 10x + 21)Step 5 x + 3–(x2 + 3x)

x

+ 21

Bring down the next term in the dividend.

Repeat Steps 2-5 as necessary.

x2 + 10x + 21)Step 6 x + 3–(x2 + 3x)

x + 7

7x + 21–(7x + 21)

0The remainder is 0.

7x


Additional Example 3A Continued

Check: Multiply the answer and the divisor.

(x + 3)(x + 7)

x2 + 7x + 3x + 21

x2 + 10x + 21


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

Helpful Hint

10-6 Dividing PolynomialsAdditional Example 3B: Polynomial Long Division


x2 – 2x – 8 )x – 4 Write in long division form.

–(x2 – 4x)2x

x2 – 2x – 8)x – 4

–(2x – 8)

0

x2 ÷ x = xMultiply x (x – 4). Subtract.

Bring down the 8. 2x ÷ x = 2.

Multiply 2(x – 4). Subtract.The remainder is 0.

x+ 2

– 8


Additional Example 3B Continued


(x + 2)(x – 4)

x2 – 4x + 2x – 8

x2 – 2x + 8

10-6 Dividing PolynomialsCheck It Out! Example 3a


(2y2 – 5y – 3) ÷ (y – 3)

2y2 – 5y – 3 )Step 1 y – 3Write in long division form

with expressions in standard form.

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

2y2 – 5y – 3)Step 2 y – 32y


Check It Out! Example 3a Continued


(2y2 – 5y – 3) ÷ (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.

2y2 – 5y – 3)Step 3 y – 32y

2y2 – 6y

–(2y2 – 6y)0 + y

2y2 – 5y – 3)Step 4 y – 3

2y




)Step 5 y – 3

2y

– 3

Bring down the next term in the dividend.

Repeat Steps 2–5 as necessary.

2y

2y2 – 5y – 3 )Step 6 y – 3–(2y2 – 6y)

y – 3 –(y – 3)

0The remainder is 0.

2y2 – 5y – 3–(2y2 – 6y)

y

+ 1



(y – 3)(2y + 1)

2y2 + y – 6y – 3

2y2 – 5y – 3




(a2 – 8a + 12) ÷ (a – 6)

a2 – 8a + 12 )a – 6 Write in long division form.

–(a2 – 6a)–2a

a a2 – 8a + 12)a – 6

–(–2a + 12)

0

a2 ÷ a = a

Multiply a (a – 6). Subtract.

Bring down the 12. –2a ÷ a = –2.

Multiply –2(a – 6). Subtract.

The remainder is 0.

– 2

+ 12


Check It Out! Example 3b Continued


(a – 6)(a – 2)

a2 – 2a – 6a + 12

a2 – 8a + 12


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.


Additional Example 4: Long Division with a Remainder

Divide (3x2 + 19x + 26) ÷ (x + 5)

3x2 + 19x + 26 )x + 5 Write in long division form.

3x2 + 19x + 26 )x + 53x

–(3x2 + 15x)4x

3x2 ÷ x = 3x.Multiply 3x(x + 5). Subtract.

Bring down the 26. 4x ÷ x = 4.

Multiply 4(x + 5). Subtract.–(4x + 20)

6 The remainder is 6.

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

+ 4

+ 26


Additional Example 4 Continued

Divide (3x2 + 19x + 26) ÷ (x + 5)

Write the quotient with the remainder.



Divide.

3m2 + 4m – 2 )m + 3 Write in long division form.

3m2 + 4m – 2 )m + 33m

–(3m2 + 9m)

3m2 ÷ m = 3m.Multiply 3m(m + 3). Subtract.

Bring down the –2. –5m ÷ m = –5 .

Multiply –5(m + 3). Subtract.–5m

The remainder is 13.13

–(–5m – 15)

– 5

– 2



Divide.

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


Divide.

y2 + 3y + 2 )y – 3 Write in long division form.

–(y2 – 3y)

y2 ÷ y = y.Multiply y(y – 3). Subtract.

Bring down the 2. 6y ÷ y = 6.

y y2 + 3y + 2 )y – 3

Multiply 6(y – 3). Subtract.

The remainder is 20.20

6y –(6y –18)

Write the quotient with the remainder.

+ 6

+ 2

y + 6 +


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.


Additional Example 5: Dividing Polynomials That Have a Zero Coefficient

Divide (x3 – 7 – 4x) ÷ (x – 3).

x3 + 0x2 – 4x – 7 )x – 3 x3 ÷ x = x2

Multiply x2(x – 3). Subtract.

(x3 – 4x – 7) ÷ (x – 3) Write the polynomials in standard form.

Write in long division form. Use 0x2 as a placeholder for the x2 term. x2

x3 + 0x2 – 4x – 7 )x – 3

–(x3 – 3x2)

3x2 – 4x Bring down –4x.


Additional Example 5 Continued

x3 + 0x2 – 4x – 7 )x – 3 3x3 ÷ x = 3xMultiply x2(x – 3). Subtract.

x2

–(x3 – 3x2)

3x2 – 4x Bring down –4x.–(3x2 – 9x)

5x–(5x – 15)

8

Bring down – 7.

Multiply 3x(x – 3). Subtract.

The remainder is 8.

+ 3x

– 7 Multiply 5(x – 3). Subtract.

+ 5

(x3 – 4x – 7) ÷ (x – 3) =


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.

Remember!


Divide (1 – 4x2 + x3) ÷ (x – 2).


(x3 – 4x2 + 1) ÷ (x – 2)

x3 – 4x2 + 0x + 1x – 2)

Write in standard form.Write in long division form.

Use 0x as a placeholder for the x term.

x3 – 4x2 + 0x + 1x – 2)x2 x3 ÷ x = x2

–(–2x2 + 4x)

– 4x –(–4x + 8)

–7

Bring down 0x. – 2x2 ÷ x = –2x.

Multiply –2x(x – 2). Subtract.Bring down 1.Multiply –4(x – 2). Subtract.

–(x3 – 2x2) – 2x2

Multiply x2(x – 2). Subtract.

– 2x

+ 0x

+ 1

– 4


Divide (1 – 4x2 + x3) ÷ (x – 2).


(1 – 4x2 + x3) ÷ (x – 2) =


Divide (4p – 1 + 2p3) ÷ (p + 1).

Check It Out! Example 5b

(2p3 + 4p – 1) ÷ (p + 1)

2p3 + 0p2 + 4p – 1p + 1)

Write in standard form.

Write in long division form. Use 0p2 as a placeholder for the p2 term.

2p3 + 0p2 + 4p – 1p + 1)2p2

p3 ÷ p = p2

–(–2p2 – 2p)

6p –(6p + 6)

–7

Bring down 4p. –2p2 ÷ p = –2p.

Multiply –2p(p + 1). Subtract.Bring down –1.Multiply 6(p + 1). Subtract.

–(2p3 + 2p2) – 2p2

Multiply 2p2(p + 1). Subtract.

– 2p

+ 4p

– 1

+ 6


Check It Out! Example 5b Continued

(2p3 + 4p – 1) ÷ (p + 1) =


Lesson Quiz: Part I

Add or Subtract. Simplify your answer.

1.

3.

2.

(12x2 – 4x2 + 20x) ÷ 4x 3x2 – x + 5

x – 2

4. x + 3

2x + 3


Lesson Quiz: Part II


5.

6. (8x2 + 2x3 + 7) (x + 3)

(x2 + 4x + 7) (x + 1)

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