division-of-polynomials

1. Review:
𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:

2. Review:
𝒂𝒙
⦁𝒂𝒚
= 𝒂𝒙+𝒚
Multiplication Rule of Exponent
𝒂𝟐+𝟑
=𝒂𝟓
𝒂𝟐
⦁𝒂𝟑
=
𝟐𝒙𝟑
⦁𝒙𝟑
=𝟐𝒙𝟑+𝟑
= 𝟐𝒙𝟔
−𝟑𝒃𝟐
⦁𝟐𝒃−𝟒
=−𝟔𝒃𝟐+(−𝟒)
=−𝟔𝒃𝟐
𝒙𝟐
𝒚𝟑
𝒙𝟐
⦁𝒚𝟑
=
𝒂−𝟏
⦁𝟐𝒂𝟑
= 𝟐𝒂−𝟏+𝟑
= 𝟐𝒂𝟐
𝟑𝒃𝟑
⦁ − 𝒃𝟒
= −𝟑𝒃𝟑+𝟒
= −𝟑𝒃𝟕
−𝟖𝒄 ⦁ − 𝟑𝒄𝟓
=−𝟖𝒄𝟏+𝟓
= −𝟐𝟒𝒄𝟔

3. Review:
𝒂𝒙
÷ 𝒂𝒚
=
𝒂𝒙
𝒂𝒚
= 𝒂𝒙−𝒚
Division Rule of Exponent
𝒂𝟓
𝒂𝟑
= 𝒂𝟓−𝟑
= 𝒂𝟐
𝟒𝒃𝟐
÷ 𝟐𝒃𝟐
=𝟐𝒃𝟐−𝟐
= 𝟐
𝟓𝒄−𝟐
−𝟏𝟎𝒄𝟑
=−𝟐𝒄−𝟐−(𝟑)
=𝟐𝒄−𝟓

5. OBJECTIVES
• Perform division of polynomials using long
division method
• Perform division of polynomials using synthetic
division.

7. Steps in dividing polynomials using long division
1. Arrange thedividend and the divisor in decreasing powers of exponents. (Note: Insert
zeros as coefficient of the missing terms of eachpolynomial if necessary.
2. Divide the first termof the dividend by the first termof the divisor.
3. Multiplythe partial quotient to the divisor.
4. Subtract the result fromthe dividend.
5. Bring down the next termin the dividend.
6. Repeat the process until done.

8. Example 1
𝐃𝐢𝐯𝐢𝐝𝐞 𝒙𝟐
+ 𝟓𝒙 + 𝟔 𝐛𝐲 𝒙 + 𝟐
1. Arrangethedividendandthedivisorin
decreasingpowersof exponents.(Note:
Insertzerosascoefficientof themissing
term)
2. Dividethefirsttermof thedividendby the
firsttermof thedivisor.
3. Multiplythepartialquotientto thedivisor.
4. Subtracttheresultfromthedividend.
5. Bringdownthenext terminthedividend.
6
5
2 2


 x
x
x
𝒙 + 𝟑
−( )
𝑥2+2𝑥
0 +3𝑥 +6
−( )
3𝑥 +6
0 +0
𝒙𝟐
+ 𝟓𝒙 + 𝟔 ÷ 𝒙 + 𝟐 = 𝒙 − 𝟑

9. Example 2
𝐃𝐢𝐯𝐢𝐝𝐞 𝒙𝟑
− 𝟒𝒙𝟐
+ 𝟑𝒙 − 𝟔 𝐛𝐲 𝒙 − 𝟐
1. Arrangethedividendandthedivisorin
decreasingpowersof exponents.(Note:
Insertzerosascoefficientof themissing
term)
2. Dividethefirsttermof thedividendby the
firsttermof thedivisor.
3. Multiplythepartialquotientto thedivisor.
4. Subtracttheresultfromthedividend.
5. Bringdownthenext terminthedividend.
6
3
4
2 2
3



 x
x
x
x
𝒙𝟐−𝟐𝒙
−( )
𝑥3−2𝑥2
0 −2𝑥2+3𝑥
−( )
−2𝑥2
+4𝑥
0 −𝑥
𝒙𝟑
− 𝟒𝒙𝟐
𝟓𝒙 + 𝟑𝒙 − 𝟔 ÷ 𝒙 − 𝟐 = 𝒙𝟐
− 𝟐𝒙 − 𝟏
−𝟏
−6
−𝑥 +2
−( )
−8

11. Steps in dividing polynomials using synthetic division
1. Write the numerical coefficients in one row. If there is a missing term, write 0 to
represent that missing term. Write the test zero at the left.
2. Bring down the leading coefficient to the bottomrow.
3. Multiplyc by the value just written on the bottomrow.
4. Add the column created in step 3.
5. Repeat until done.
6. Writeout the answer.

12. Example 1.)

13. Example 1.)

14. Example 1.)