Polynomial Division: Dividing a Polynomial by a Monomial and a Polynomial, Study notes of Algebra

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Objectives:

• Divide a polynomial by a monomial.
• Divide a polynomial by a polynomial of two or more terms.
• Divide polynomial functions.

5.5 Dividing Polynomials

Written by: Cindy Alder

Parts of a Division Problem

 There are three parts to a division problem: the dividend , the divisor , and the quotient.

 A division problem can be written three different ways:

𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 ÷ 𝑑𝑖𝑣𝑖𝑠𝑜𝑟 = 𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡

quotient

divisor dividend

Example 1

 Divide.

50𝑚^4 − 30𝑚^3 + 20𝑚 10𝑚^3

Example 2

 Divide.

(−8𝑝^4 − 6𝑝^3 − 12𝑝^5 ) ÷ (−3𝑝^3 )

Long Division

 Divide. 3,257 ÷ 12

 Step 1: Rewrite as a long division problem. Make sure that both polynomials are written in descending order, fill in any missing terms with a zero term.

 Step 2:

a) Take the first term of the dividend and divide by the first term of the divisor:

Place this value above it’s like term.

Divide: 𝟓𝒙 − 𝟖 + 𝟒𝒙𝟑^ − 𝟒𝒙𝟐^ ÷ (𝟐𝒙 − 𝟏)

 Step 3: Bring down the next term.

 Step 4: Repeat steps 2 and 3 until you have brought down the last term.

Dividing a Polynomial by a Polynomial

 Step 5: State your answer. If there is a remainder,

place the remainder over the divisor and add it to

the quotient.

Dividing a Polynomial by a Polynomial

Dividing a Polynomial by a Polynomial

To divide a polynomial by a polynomial, follow the six steps outlined below.

 Step 1 : Rewrite as a long division problem. Make sure that both polynomials are written in descending order, filling in any missing terms with a zero term.  Step 2 : a) Divide the first term of the dividend by the first term of the divisor. Place that quotient above it’s like term. b) Multiply the quotient from part a by the divisor. Place that product below it’s like term. c) Subtract.  Step 3 : Bring down the next term.  Step 4 : Repeat steps 2 and 3 until you have brought down the last term.  Step 5 : State your answer. If there is a remainder, place the remainder over the divisor and add it to the quotient.  Step 6 : Check. Multiply the divisor by the quotient and add the remainder.

Example 4

 Divide 8𝑥^3 − 4𝑥^2 − 14𝑥 + 15 𝑏𝑦 2𝑥 + 3.

Example 6

 Divide 4𝑥^3 + 3𝑥 − 8 𝑏𝑦 𝑥 + 2



Example 7

 Divide

6𝑚^4 + 9𝑚^3 + 2𝑚^2 − 8𝑚 + 7 3𝑚^2 − 2

Example 9

 For 𝑓 𝑥 = 2𝑥^2 + 17𝑥 + 30 and 𝑔 𝑥 = 2𝑥 + 5,

find

𝑓

𝑔 (𝑥)^ and^

𝑓