Division of Polynomials

Digital Lesson

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Dividing Polynomials

Long division of polynomials is similar to long division of whole numbers.

dividend = (quotient • divisor) + remainder

The result is written in the form:

quotient +divisor

remainder divisor dividend

When you divide two polynomials you can check the answer using the following:

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+ 2 2 3 1 2 xxx

Example: Divide x2 + 3x – 2 by x – 1 and check the answer.

x

x2 + x2x – 22x + 2

– 4

remainder

Check:

xx

xxx

22 1.

xxxx 2)1(2.

xxxxx 2)()3( 22 3.

22

2 x

xxx4.

22)1(2 xx5.

4)22()22( xx6.

correct(x + 2)

quotient

(x + 1)

divisor

+ (– 4)

remainder

= x2 + 3x – 2

dividend

Answer: x + 2 +1x

– 4

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Example: Divide 4x + 2x3 – 1 by 2x – 2 and check the answer.

1 4 0 2 2 2 23 xxxx Write the terms of the dividend in

descending order.

23

2

2x

x

x1.

x2

232 22)22( xxxx 2.

2x3 – 2x2

2233 2)22(2 xxxx 3.

2x2 + 4x

xx

x

2

2 2

4.

+ x

xxxx 22)22( 2 5.

2x2 – 2x

xxxxx 6)22()42( 22 6.

6x – 1

32

6

x

x7.

+ 3

66)22(3 xx8.

6x – 6

remainder5)66()16( xx9.

5

Check: (x2 + x + 3)(2x – 2) + 5

= 4x + 2x3 – 1

Answer: x2 + x + 322

x5

Since there is no x2 term in the

dividend, add 0x2 as a placeholder.

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6 5 2 2 xxxx

x2 – 2x

– 3x + 6

– 3

– 3x + 60

Answer: x – 3 with no remainder.

Check: (x – 2)(x – 3) = x2 – 5x + 6

Example: Divide x2 – 5x + 6 by x – 2.

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Example: Divide x3 + 3x2 – 2x + 2 by x + 3 and check the answer.

2 2 3 3 23 xxxx

x2

x3 + 3x2

0x2 – 2x

– 2

– 2x – 6

8

Check: (x + 3)(x2 – 2) + 8

= x3 + 3x2 – 2x + 2

Answer: x2 – 2 +3x

8

+ 2

Note: the first subtraction

eliminated two terms from

the dividend.

Therefore, the quotient

skips a term.

+ 0x