Polynomial Long Division Copyright © 1999 Lynda Greene

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

? 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

- 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)

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

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

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

SOLUTION: Rearrange terms into descending order Polynomial Long Division PROBLEM: Terms out of descending order Divide : 3x5 - 17x4 - 15x3 + 4x + 54x2 - 24 by x - 6 SOLUTION: Rearrange terms into descending order

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

? 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

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

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

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

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 + 9x3 - 54x2 Be sure to change the signs of every term. 0x2 + 4x

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 + 9x3 - 54x2 Subtract (The first terms should always cancel out) + 0x2 + 4x Be sure to change the signs of every term. - 0x2 + 0x + 4x - 24

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 + 9x3 - 54x2 + 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

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.

Watch out for missing powers! Polynomial Long Division Divide : 6x4 - 3x3 - 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!

SOLUTION: Insert the missing power with a zero coefficient Polynomial Long Division Divide : 6x4 - 3x3 - x - 5 by 2x - 3 PROBLEM: Missing the x2 term SOLUTION: Insert the missing power with a zero coefficient

Polynomial Long Division Divide : 6x4 - 3x3 - 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

? 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

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

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

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

+ 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

+ 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

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

Practice Problems: Divide using Polynomial Long Division Answers: 1) 5x - 1 2) 6x + 5 3) 4x2 + 7x + 12 + 4) x2 - 2x - 7 5) 5x2 + 10x + 22 + 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

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