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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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