1. Polynomial Long Division Copyright © 1999 Lynda Greene

2. Polynomial Long Division
Divide : 2x3 + 5x2 - x + 6 by x + 3
Step 1: Write the problem using a division symbol
Step 2: Look at the first term on the outside and the inside

3. ? 2x2 So, the term we are looking for is 2x2
Step 3:The outside term (x) was multiplied by (something) to equal
(2x3), the inside term. We must figure out what that (something) was.
x times (what?) = 2x3
2x2 ?
Put 2x2 on
the top
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.
So, the term we are looking for is 2x2

4. - 2x2 2x3 + 6x2 2x2(x + 3) = 2x3 + 6x2 = -2x3- 6x2 Multiply the term
you just wrote on
top by the outside
terms.
2x2(x + 3)
= 2x3 + 6x2
Be sure to change the signs of every term. -
2x3 + 6x2
The next step is subtraction so we have: -(2x3 + 6x2)
= -2x3- 6x2
(This answer will be written in the next line, under the correct powers)

5. 2x2 - x - x -2x3 - 6x2 - x2 Subtract Bring down the next term
(The first terms
should always
cancel out)
-2x3 - 6x2
Bring down
the next term
- x
- x2
Step 2: what did we multiply the outside term by to get the inside term.
Step 3: Write this term on top
Now we will repeat the whole process again.
Step 1: look at the first terms

6. 2x2 - x - x -2x3 - 6x2 - x2 + x2 + 3x + 2x + 6 Bring down
the next term
-2x3 - 6x2
Subtract
(The first terms
should always
cancel out)
- x
- x2
Be sure to change the signs of every term.
+ x2 + 3x
+ 2x
+ 6
Step 4: Multiply this new term by the outside terms
Step 5: Change the signs & write the answer under the current inside term

7. 2x2 - x + 2 - x -2x3 - 6x2 - x2 + x2 + 3x + 2x + 6 - 2x - 6
ANSWER IS ON TOP
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
2x2
- x
+ 2
-2x3 - 6x2
- x
- x2
+ x2 + 3x
Subtract
(The first terms
should always
cancel out)
Be sure to change the signs of every term.
+ 2x
+ 6
- 2x - 6

8. SOLUTION: Rearrange terms into descending order
Polynomial Long Division
PROBLEM: Terms out of descending order
Divide : 3x5 - 17x4 - 15x3 + 4x + 54x by x - 6
SOLUTION: Rearrange terms into descending order

9. Polynomial Long Division
Divide : 3x5 - 17x4 - 15x3 + 54x2 + 4x - 24 by x - 6
Step 1: Write the problem using a division symbol
Step 2: Look at the first term on the outside and the inside

10. ? 3x4 So, the term we are looking for is 3x4
Step 3:The outside term (x) was multiplied by (something) to equal
(3x5), the inside term. We must figure out what that (something) was.
x times (what?) = 3x5
Put 3x4 on
the top
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.
So, the term we are looking for is 3x4

11. Multiply the term you just wrote on top by the outside terms.
3x4(x - 6)
= 3x5 - 18x4
3x4
Be sure to change the signs of every term.
3x x4
The next step is subtraction so we have: -(3x5 - 18x4)
= -3x5+ 18x4

12. 3x4 + x3 - 15x3 -3x5+ 18x4 + x4 Subtract Bring down the next term
(The first terms
should always
cancel out)
-3x5+ 18x4
Bring down
the next term
- 15x3
+ x4
Step 2: what did we multiply the outside term by to get the inside term.
Step 3: Write this term on top
Now we will repeat the whole process again.
Step 1: look at the first terms

13. 3x4 + x3 -3x5+ 18x4 + x4 - 15x3 - x4 + 6x3 - 9x3 + 54x2 Bring down
Step 4: Multiply this new term by the outside terms
Step 5: Change the signs & write the answer under the current inside term
3x4
+ x3
-3x5+ 18x4
Bring down
the next term
Subtract
(The first terms
should always
cancel out)
+ x4
- 15x3
- x4 + 6x3
Be sure to change the signs of every term.
- 9x3
+ 54x2

14. 3x4 + x3 - 9x2 -3x5+ 18x4 + x4 - 15x3 - x4 + 6x3 - 9x3 + 54x2
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
3x4
+ x3
- 9x2
-3x5+ 18x4
Bring down
the next term
+ x4
- 15x3
- x4 + 6x3
Subtract
(The first terms
should always
cancel out)
- 9x3
+ 54x2
+ 9x x2
Be sure to change the signs of every term.
0x2
+ 4x

15. 3x4 + x3 - 9x2 + 0x -3x5+ 18x4 + x4 - 15x3 - x4 + 6x3 - 9x3 + 54x2
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
3x4
+ x3
- 9x2
+ 0x
-3x5+ 18x4
Bring down
the next term
+ x4
- 15x3
- x4 + 6x3
- 9x3
+ 54x2
+ 9x x2
Subtract
(The first terms
should always
cancel out)
+ 0x2
+ 4x
Be sure to change the signs of every term.
- 0x2 + 0x
+ 4x
- 24

16. 3x4 + x3 - 9x2 + 0x + 4 -3x5+ 18x4 + x4 - 15x3 - x4 + 6x3 - 9x3 + 54x2
ANSWER IS ON TOP
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
3x4
+ x3
- 9x2
+ 0x
+ 4
-3x5+ 18x4
+ x4
- 15x3
- x4 + 6x3
- 9x3
+ 54x2
+ 9x x2
+ 0x2
+ 4x
Subtract
(The first terms
should always
cancel out)
- 0x2 + 0x
+ 4x
- 24
Be sure to change the signs of every term.
- 4x + 24

17. The division problems we just worked ended with zero remainders.
Now let’s work a problem that ends with a remainder that’s not a zero.
We’ll also throw in a couple of fractions so you can see how they are handled.

18. Watch out for missing powers!
Polynomial Long Division
Divide : 6x4 - 3x x - 5 by 2x - 3
Step 1: Write the problem using a division symbol
This polynomial (inside) has a power missing (x2). This is a common occurrence in polynomial long division problems.
Watch out for missing powers!

19. SOLUTION: Insert the missing power with a zero coefficient
Polynomial Long Division
Divide : 6x4 - 3x x - 5 by 2x - 3
PROBLEM: Missing the x2 term
SOLUTION: Insert the missing power with a zero coefficient

20. Polynomial Long Division
Divide : 6x4 - 3x x - 5 by 2x - 3
Step 1: Write the problem using a division symbol
Step 2: Look at the first term on the outside and the inside

21. ? 3x3 So, the term we are looking for is 3x3
Step 3:The outside term (x) was multiplied by (something) to equal
(6x4), the inside term. We must figure out what that (something) was.
x times (what?) = 6x4
3x3 ?
Put 3x3 on
the top
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.
So, the term we are looking for is 3x3

22. Be sure to change the signs of every term.
Multiply the term
you just wrote on
top by the outside
terms.
3x3(2x - 3)
= 6x4 - 9x3
3x3
Be sure to change the signs of every term.
6x x3
The next step is subtraction so we have: -(6x4 - 9x3)
= - 6x4 + 9x3

23. 3x3 + 3x2 + 0x2 -6x4 + 9x3 6x3 Subtract Bring down the next term
(The first terms
should always
cancel out)
-6x4 + 9x3
Bring down
the next term
+ 0x2
6x3
Step 2: what did we multiply the outside term by to get the inside term.
Step 3: Write this term on top
Now we will repeat the whole process again.
Step 1: look at the first terms

24. 3x3 + 3x2 -6x4 + 9x3 6x3 + 0x2 - 6x3 + 9x2 - x + 9x2 Bring down
Step 4: Multiply this new term by the outside terms
Step 5: Change the signs & write the answer under the current inside term
3x3
+ 3x2
-6x4 + 9x3
Bring down
the next term
Subtract
(The first terms
should always
cancel out)
6x3
+ 0x2
Be sure to change the signs of every term.
- 6x3 + 9x2
- x 17 2
+ 9x2

25. + x 3x3 + 3x2 -6x4 + 9x3 6x3 + 0x2 - 6x3 + 9x2 - x + 9x2 - 9x2+ x 5x
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
+ x 9 2
3x3
+ 3x2
-6x4 + 9x3
Bring down
the next term
6x3
+ 0x2
- 6x3 + 9x2
Subtract
(The first terms
should always
cancel out)
Be sure to change the signs of every term.
- x 17 2
+ 9x2
- 9x2+ x 27 2 5x
- 5
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

26. + 3x3 + 3x2 + x + -6x4 + 9x3 6x3 + 0x2 - 6x3 + 9x2 - x + 9x2 - 9x2+ x
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
ANSWER IS ON TOP +
5/2
2x-3
The remainder is written as a fraction. the remainder over the divisor (outside polynomial)
REMAINDER
DIVISOR 9 2 5 2 +
3x3
+ 3x2
+ x
-6x4 + 9x3
Be sure to change the sign of every term.
6x3
+ 0x2
- 6x3 + 9x2
- x 17 2
+ 9x2
- 9x2+ x 27 2
Subtract
(The first terms
should always
cancel out) 5x
- 5
- 5x + 15 2
5/2
No more terms to bring down,
this (5/2) is the remainder

27. Practice Problems: 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

28. Practice Problems: Divide using Polynomial Long Division
Answers:
1) 5x - 1
2) 6x + 5
3) 4x2 + 7x
4) x2 - 2x - 7
5) 5x2 + 10x
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

29. Questions? send e-mail to: lgreene1@satx.rr.com
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