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The process was to divide, multiply, and subtract.

Review of subtraction of polynomials

A very important step in the process of dividing one polynomial by another is subtraction of polynomials. Let’s review the process of subtraction by observing a few examples.

1. Subtract x 2 from x 5 ; that is, find ( x 5 ) ( x 2 ) .

 Since   x 2 is preceded by a minus sign, remove the parentheses, change the sign of each term, then add.

  x 5 ( x 2 ) = x 5 x + 2 3

The result is 3.

2. Subtract x 3 + 3 x 2 from x 3 + 4 x 2 + x 1.

 Since x 3 + 3 x 2 is preceded by a minus sign, remove the parentheses, change the sign of each term, then add.

x 3 + 4 x 2 + x 1 ( x 3 + 3 x 2 ) = x 3 + 4 x 2 + x 1 x 3 3 x 2 x 2 + x 1

The result is x 2 + x 1.

3. Subtract x 2 + 3 x from x 2 + 1.

 We can write x 2 + 1 as x 2 + 0 x + 1.

x 2 + 1 ( x 2 + 3 x ) = x 2 + 0 x + 1 ( x 2 + 3 x ) = x 2 + 0 x + 1 x 2 3 x 3 x + 1

Dividing a polynomial by a polynomial

Now we’ll observe some examples of dividing one polynomial by another. The process is the same as the process used with whole numbers: divide, multiply, subtract, divide, multiply, subtract,....

The division, multiplication, and subtraction take place one term at a time. The process is concluded when the polynomial remainder is of lesser degree than the polynomial divisor.

Sample set b

Perform the division.

x 5 x 2 . We are to divide  x 5  by  x 2.

Long division showing x minus two dividing x minus five with the comment 'Divide x into x' on the right side. This division is not performed completely. See the longdesc for a full description.

1 3 x 2 Thus, x 5 x 2 = 1 3 x 2

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x 3 + 4 x 2 + x 1 x + 3 . We are to divide  x 3 + 4 x 2 + x 1  by  x + 3.

Long division showing x plus three dividing x cube plus four x square plus x minus one with the comment 'Divide x into x cube' on the right side. This division is not performed completely. See the longdesc for a full description

x 2 + x 2 + 5 x + 3 Thus, x 3 + 4 x 2 + x 1 x + 3 = x 2 + x 2 + 5 x + 3

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Practice set b

Perform the following divisions.

x + 6 x 1

1 + 7 x 1

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x 2 + 2 x + 5 x + 3

x 1 + 8 x + 3

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x 3 + x 2 x 2 x + 8

x 2 7 x + 55 442 x + 8

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x 3 + x 2 3 x + 1 x 2 + 4 x 5

x 3 + 14 x 14 x 2 + 4 x 5 = x 3 + 14 x + 5

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Sample set c

Divide  2 x 3 4 x + 1 by  x + 6. 2 x 3 4 x + 1 x + 6 Notice that the  x 2  term in the numerator is missing .  We can avoid any confusion by writing 2 x 3 + 0 x 2 4 x + 1 x + 6 Divide, multiply, and subtract .

Steps of long division showing the quantity x plus six dividing the quantity two x cubed plus zero x squared minus four x minus plus one. See the longdesc for a full description

2 x 3 4 x + 1 x + 6 = 2 x 3 12 x + 68 407 x + 6

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Practice set c

Perform the following divisions.

x 2 3 x + 2

x 2 + 1 x + 2

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4 x 2 1 x 3

4 x + 12 + 35 x 3

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x 3 + 2 x + 2 x 2

x 2 + 2 x + 6 + 14 x 2

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6 x 3 + 5 x 2 1 2 x + 3

3 x 2 2 x + 3 10 2 x + 3

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Exercises

For the following problems, perform the divisions.

8 y 4 4

2 y + 1

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3 x 2 6 x 3

x ( x 2 )

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6 x 3 + 2 x 2 + 8 x 2 x

3 x 2 + x + 4

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26 y 3 + 13 y 2 + 39 y 13 y

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a 2 b 2 + 4 a 2 b + 6 a b 2 10 a b a b

a b + 4 a + 6 b 10

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7 x 3 y + 8 x 2 y 3 + 3 x y 4 4 x y x y

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5 x 3 y 3 15 x 2 y 2 + 20 x y 5 x y

x 2 y 2 + 3 x y 4

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4 a 2 b 3 8 a b 4 + 12 a b 2 2 a b 2

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6 a 2 y 2 + 12 a 2 y + 18 a 2 24 a 2

1 4 y 2 + 1 2 y + 3 4

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3 c 3 y 3 + 99 c 3 y 4 12 c 3 y 5 3 c 3 y 3

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16 a x 2 20 a x 3 + 24 a x 4 6 a 4

8 x 2 10 x 3 + 12 x 4 3 a 3 or 12 x 4 10 x 3 + 8 x 2 3 a 3

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21 a y 3 18 a y 2 15 a y 6 a y 2

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14 b 2 c 2 + 21 b 3 c 3 28 c 3 7 a 2 c 3

2 b 2 3 b 3 c + 4 c a 2 c

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30 a 2 b 4 35 a 2 b 3 25 a 2 5 b 3

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x + 6 x 2

1 + 8 x 2

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x 2 x + 4 x + 2

x 3 + 10 x + 2

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x 2 x + 3 x + 1

x 2 + 5 x + 1

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x 2 2 x + 1

x 1 1 x + 1

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y 2 + 4 y + 2

y 2 + 8 y + 2

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x 3 1 x + 1

x 2 x + 1 2 x + 1

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x 3 1 x 1

x 2 + x + 1

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x 3 + 3 x 2 + x 2 x 2

x 2 + 5 x + 11 + 20 x 2

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a 3 + 2 a 2 a + 1 a 3

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a 3 + a + 6 a 1

a 2 + a + 2 + 8 a 1

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y 3 + 3 y 2 + 4 y + 2

y 2 + y 2 + 8 y + 2

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y 3 + 5 y 2 3 y 1

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x 2 x 6 x 2 2 x 3

1 + 1 x + 1

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a 2 + 5 a + 4 a 2 a 2

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2 y 2 + 5 y + 3 y 2 3 y 4

2 + 11 y 4

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3 a 2 + 4 a 4 a 2 + 3 a + 3

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2 x 2 x + 4 2 x 1

x + 4 2 x 1

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6 x 2 + 8 x 1 3 x + 4

2 x 1 3 x + 4

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20 y 2 + 15 y 4 4 y + 3

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4 x 3 + 4 x 2 3 x 2 2 x 1

2 x 2 + 3 x 2 2 x 1

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9 a 3 18 a 2 + 8 a 1 3 a 2

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4 x 4 4 x 3 + 2 x 2 2 x 1 x 1

4 x 3 + 2 x 1 x 1

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3 y 4 + 9 y 3 2 y 2 6 y + 4 y + 3

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3 y 2 + 3 y + 5 y 2 + y + 1

3 + 2 y 2 + y + 1

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2 a 2 + 4 a + 1 a 2 + 2 a + 3

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8 z 6 4 z 5 8 z 4 + 8 z 3 + 3 z 2 14 z 2 z 3

4 z 5 + 4 z 4 + 2 z 3 + 7 z 2 + 12 z + 11 + 33 2 z 3

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9 a 7 + 15 a 6 + 4 a 5 3 a 4 a 3 + 12 a 2 + a 5 3 a + 1

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( 2 x 5 + 5 x 4 1 ) ÷ ( 2 x + 5 )

x 4 1 2 x + 5

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( 6 a 4 2 a 3 3 a 2 + a + 4 ) ÷ ( 3 a 1 )

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Exercises for review

( [link] ) Find the product. x 2 + 2 x 8 x 2 9 · 2 x + 6 4 x 8 .

x + 4 2 ( x 3 )

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( [link] ) Find the sum. x 7 x + 5 + x + 4 x 2 .

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( [link] ) Solve the equation 1 x + 3 + 1 x 3 = 1 x 2 9 .

x = 1 2

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( [link] ) When the same number is subtracted from both the numerator and denominator of 3 10 , the result is 1 8 . What is the number that is subtracted?

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( [link] ) Simplify 1 x + 5 4 x 2 25 .

x 5 4

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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