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Polynomial Long

Division

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Divide : 2x3 + 5x2 - x + 6 by x + 3Step 1: Write the problem using a division symbol

6523 23 xxxx

Step 2: Look at the first term on the outside and the inside

Polynomial Long DivisionPolynomial Long Division

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x times (what?) = 2x3

Step 3:The outside term (x) was multiplied by (something) to equal(2x3), the inside term. We must figure out what that (something) was.

? Put 2x2 onthe top

6523 23 xxxx

So, the term we are looking for is 2x2

Well, we started with one x and we ended up with x3, so we picked up two more x’s or x2.

Also, we now have a 2 that we didn’t have before.

2x2

5

2x2

6523 23 xxxx 2x3 + 6x2

Be sure to change the signs of every term.

The next step is subtraction so we have: -(2x3 + 6x2)

= -2x3- 6x2

Multiply the termyou just wrote on top by the outsideterms.2x2(x + 3) = 2x3 + 6x2

(This answer will be written in (This answer will be written in the next line, under the correct the next line, under the correct powers)powers)

-

6

2x2 - x

- x2 - x

6523 23 xxxx-2x3 - 6x2

Now we will repeat the whole process again.Step 1: look at the first terms

Subtract(The first termsshould alwayscancel out) Bring down

the next term

Step 2: what did we multiply the outside term by to get the inside term.

Step 3: Write this term on top

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2x2

+ 6

- x

- x2 - x

+ 2x

6523 23 xxxx-2x3 - 6x2

+ x2 + 3x Be sure to change the signs of every term.

Bring downthe next term

Step 4: Multiply this new term by the outside terms

Step 5: Change the signs & write the answer under the current inside term

Subtract(The first termsshould alwayscancel out)

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ANSWER IS ON TOP

2x2

+ 6

0

- x + 2

- x2 - x

+ 2x

6523 23 xxxx-2x3 - 6x2

+ x2 + 3x

- 2x - 6 Be sure to change the signs of every term.

Repeat Steps 1-5

Step 1: Look at first terms

Step 2: What did we multiply by?

Step 3: Write this above the line

Step 4: Multiply new term by outside terms

Step 5: Change signs & subtract

Subtract(The first termsshould alwayscancel out)

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PROBLEM: Terms out of descending order

Divide : 3x5 - 17x4 - 15x3 + 4x + 54x2 - 24 by x - 6

Polynomial Long DivisionPolynomial Long Division

24454151736 2345 xxxxxx

SOLUTION: Rearrange terms into descending order

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Step 1: Write the problem using a division symbol

Step 2: Look at the first term on the outside and the inside

Polynomial Long DivisionPolynomial Long Division

24454151736 2345 xxxxxx

Divide : 3x5 - 17x4 - 15x3 + 54x2 + 4x - 24 by x - 6

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x times (what?) = 3x5

Step 3:The outside term (x) was multiplied by (something) to equal(3x5), the inside term. We must figure out what that (something) was.

?Put 3x4 on

the top

So, the term we are looking for is 3x4

Well, we started with one x and we ended up with x5, so we picked up four more x’s or x4.

Also, we now have a 3 that we didn’t have before.

3x4

24454151736 2345 xxxxxx

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3x4

Be sure to change the signs of every term.

The next step is subtraction so we have: -(3x5 - 18x4)

= -3x5+ 18x4

Multiply the term you just wrote on top by the outside terms.

3x4(x - 6) = 3x5 - 18x4

24454151736 2345 xxxxxx3x5 - 18x4- +

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3x4 + x3

+ x4 - 15x3

Now we will repeat the whole process again.Step 1: look at the first terms

Subtract(The first termsshould alwayscancel out) Bring down

the next term

Step 2: what did we multiply the outside term by to get the inside term.

Step 3: Write this term on top

24454151736 2345 xxxxxx-3x5+ 18x4

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+ 54x2- 9x3

- x4 + 6x3 Be sure to change the signs of every term.

Bring downthe next term

Step 4: Multiply this new term by the outside terms

Step 5: Change the signs & write the answer under the current inside term

Subtract(The first termsshould alwayscancel out)

24454151736 2345 xxxxxx

3x4 + x3

+ x4 - 15x3 -3x5+ 18x4

15

0x2

- 9x2

- 9x3 - 54x2

Be sure to change the signs of every term.

Repeat Steps 1-5

Step 1: Look at first terms

Step 2: What did we multiply by?

Step 3: Write this above the line

Step 4: Multiply new term by outside terms

Step 5: Change signs & subtract

Subtract(The first termsshould alwayscancel out)

24454151736 2345 xxxxxx

+ 54x2- 9x3

- x4 + 6x3

-3x5+ 18x4

+ x4 - 15x3

3x4 + x3

+ 4x

Bring downthe next term

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+ 0x2

- 9x2

- 9x3 - 54x2

Be sure to change the signs of every term.

Repeat Steps 1-5

Step 1: Look at first terms

Step 2: What did we multiply by?

Step 3: Write this above the line

Step 4: Multiply new term by outside terms

Step 5: Change signs & subtract

Subtract(The first termsshould alwayscancel out)

24454151736 2345 xxxxxx

+ 54x2- 9x3

- x4 + 6x3

-3x5+ 18x4

+ x4 - 15x3

3x4 + x3

+ 4x

Bring downthe next term

+ 0x

- 0x2 + 0x+ 4x - 24

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+ 0x2

- 9x2

- 9x3 - 54x2

Be sure to change the signs of every term.

Repeat Steps 1-5

Step 1: Look at first terms

Step 2: What did we multiply by?

Step 3: Write this above the line

Step 4: Multiply new term by outside terms

Step 5: Change signs & subtract Subtract

(The first termsshould alwayscancel out)

24454151736 2345 xxxxxx

+ 54x2- 9x3

- x4 + 6x3

-3x5+ 18x4

+ x4 - 15x3

3x4 + x3

+ 4x

+ 0x

- 0x2 + 0x+ 4x - 24

+ 4

- 4x + 240

ANSWER IS ON TOP

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Now let’s work a problem that Now let’s work a problem that ends with a remainder that’s not ends with a remainder that’s not

a zero. a zero.

We’ll also throw in a couple of We’ll also throw in a couple of fractions so you can see how fractions so you can see how

they are handled.they are handled.

The division problems we just The division problems we just worked ended with zeroworked ended with zero

remainders.remainders.

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Step 1: Write the problem using a division symbol

Polynomial Long DivisionPolynomial Long Division

This polynomial (inside) has a power missing (x2). This is a common occurrence in polynomial long

division problems.

Watch out for missing powers!

53632 21734 xxxx

Divide : 6x4 - 3x3 - x - 5 by 2x - 3217

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Polynomial Long DivisionPolynomial Long Division

SOLUTION: Insert the missing power with a zero coefficient

503632 217234 xxxxx

Divide : 6x4 - 3x3 - x - 5 by 2x - 3217

PROBLEM: Missing the x2 term

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Step 1: Write the problem using a division symbol

Step 2: Look at the first term on the outside and the inside

Polynomial Long DivisionPolynomial Long Division

503632 217234 xxxxx

Divide : 6x4 - 3x3 - x - 5 by 2x - 3217

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x times (what?) = 6x4

Step 3:The outside term (x) was multiplied by (something) to equal(6x4), the inside term. We must figure out what that (something) was.

? Put 3x3 onthe top

So, the term we are looking for is 3x3

Well, we started with one x and we ended up with x4, so we picked up three more x’s or x3.

Also, the 2 changed into a 6, so we multiplied by 3.

3x3

503632 217234 xxxxx

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3x3

6x4 - 9x3 Be sure to change the

signs of every term.

The next step is subtraction so we have: -(6x4 - 9x3)

= - 6x4 + 9x3

Multiply the termyou just wrote on top by the outsideterms.3x3(2x - 3) = 6x4 - 9x3

- + 503632 2

17234 xxxxx

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3x3 + 3x2

6x3 + 0x2 -6x4 + 9x3

Now we will repeat the whole process again.Step 1: look at the first terms

Subtract(The first termsshould alwayscancel out) Bring down

the next term

Step 2: what did we multiply the outside term by to get the inside term.

Step 3: Write this term on top

503632 217234 xxxxx

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3x3 + 3x2

6x3+ 0x2

+ 9x2

- 6x3 + 9x2 Be sure to change the signs of every term.

Bring downthe next term

Step 4: Multiply this new term by the outside terms

Step 5: Change the signs & write the answer under the current inside term

Subtract(The first termsshould alwayscancel out)

-6x4 + 9x3

503632 217234 xxxxx

- x172

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3x3 + 3x

6x3 + 0x2

+ 9x2

- 6x3 + 9x2 Be sure to change the signs of every term.

Repeat Steps 1-5

Step 1: Look at first terms

Step 2: What did we multiply by?

Step 3: Write this above the line

Step 4: Multiply new term by outside terms

Step 5: Change signs & subtract

Subtract(The first termsshould alwayscancel out)

503632 217234 xxxxx

-6x4 + 9x3 Bring downthe next term

If the coefficient of the outside term, 2x, does not go evenly into the coefficient of the inside term, 9x2,

then the number that goes on top will be: (inside/outside)= 9/2

+ x92

- 9x2+ x 272

5x

- x172

- 5

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ANSWER IS ON TOP

3x3 + 3x

6x3 + 0x2

+ 9x2

- 6x3 + 9x2

Be sure to change the sign of every term.

Repeat Steps 1-5

Step 1: Look at first terms

Step 2: What did we multiply by?

Step 3: Write this above the line

Step 4: Multiply new term by outside terms

Step 5: Change signs & subtract

Subtract(The first termsshould alwayscancel out)

-6x4 + 9x3

+ x503632 2

17234 xxxxx

92

- 9x2+ x 272

5x

- x172

- 5- 5x + 15

2

52+

5No more terms to bring down, this (5) is the remainder

+ 52x-3

The remainder is written as a fraction. the remainder over the divisor (outside polynomial)

REMAINDER

DIVISOR

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Practice Problems: (Hit enter to see the answers)Divide using Polynomial Long Division

1) 15x2 + 22x - 5 by 3x + 5

2) 12x2 - 32x - 35 by 2x - 7

3) 4x3 - 2x - x2 + 6 by x - 2

4) 3x3 - 5x2 - 23x - 7 by 3x + 1

5) 5x3 + 2x - 3 by x - 2

Answers:1) 5x - 12) 6x + 53) 4x2 + 7x + 12 +

4) x2 - 2x - 75) 5x2 + 10x + 22 +

2

30

x

2

41

x

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