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2 Polynomial Long Division

3 Divide : 2x 3 + 5x 2 - 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 Polynomial Long Division

4 x times (what?) = 2x 3 Step 3:The outside term (x) was multiplied by (something) to equal (2x 3 ), the inside term. We must figure out what that (something) was. ? Put 2x 2 on the top So, the term we are looking for is 2x 2 Well, we started with one x and we ended up with x 3, so we picked up two more x’s or x 2. Also, we now have a 2 that we didn’t have before. 2x 2

5 2x 3 + 6x 2 Be sure to change the signs of every term. The next step is subtraction so we have: -( 2x 3 + 6x 2 ) = -2x 3 - 6x 2 Multiply the term you just wrote on top by the outside terms. 2x 2 (x + 3) = 2x 3 + 6x 2 (This answer will be written in the next line, under the correct powers) -

6 2x 2 - x - x 2 - x -2x 3 - 6x 2 Now we will repeat the whole process again. Step 1: look at the first terms Subtract (The first terms should always cancel 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

7 2x x - x 2 - x + 2x -2x 3 - 6x 2 + x 2 + 3x Be sure to change the signs of every term. Bring down the 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 terms should always cancel out)

8 ANSWER IS ON TOP 2x x+ 2 - x 2 - x + 2x -2x 3 - 6x 2 + x 2 + 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 terms should always cancel out)

9 PROBLEM: Terms out of descending order Divide : 3x x x 3 + 4x + 54x by x - 6 Polynomial Long Division SOLUTION: Rearrange terms into descending order

10 Step 1: Write the problem using a division symbol Step 2: Look at the first term on the outside and the inside Polynomial Long Division Divide : 3x x x x 2 + 4x - 24 by x - 6

11 x times (what?) = 3x 5 Step 3:The outside term (x) was multiplied by (something) to equal (3x 5 ), the inside term. We must figure out what that (something) was. ? Put 3x 4 on the top So, the term we are looking for is 3x 4 Well, we started with one x and we ended up with x 5, so we picked up four more x’s or x 4. Also, we now have a 3 that we didn’t have before. 3x 4

12 3x 4 Be sure to change the signs of every term. The next step is subtraction so we have: -(3 x x 4 ) = -3x x 4 Multiply the term you just wrote on top by the outside terms. 3x 4 (x - 6) = 3x x 4 3x x 4 - +

13 3x 4 + x 3 + x x 3 Now we will repeat the whole process again. Step 1: look at the first terms Subtract (The first terms should always cancel 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 -3x x 4

x 2 - 9x 3 - x 4 + 6x 3 Be sure to change the signs of every term. Bring down the 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 terms should always cancel out) 3x 4 + x 3 + x x 3 -3x x 4

15 0x 2 - 9x 2 - 9x x 2 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 terms should always cancel out) + 54x 2 - 9x 3 - x 4 + 6x 3 -3x x 4 + x x 3 3x 4 + x 3 + 4x Bring down the next term

16 + 0x 2 - 9x 2 - 9x x 2 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 terms should always cancel out) + 54x 2 - 9x 3 - x 4 + 6x 3 -3x x 4 + x x 3 3x 4 + x 3 + 4x Bring down the next term + 0x - 0x 2 + 0x + 4x- 24

17 + 0x 2 - 9x 2 - 9x x 2 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 terms should always cancel out) + 54x 2 - 9x 3 - x 4 + 6x 3 -3x x 4 + x x 3 3x 4 + x 3 + 4x + 0x - 0x 2 + 0x + 4x x ANSWER IS ON TOP

18 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. The division problems we just worked ended with zero remainders. remainders.

19 Step 1: Write the problem using a division symbol Polynomial Long Division This polynomial (inside) has a power missing (x 2 ). This is a common occurrence in polynomial long division problems. Watch out for missing powers! Divide : 6x 4 - 3x 3 - x - 5 by 2x - 3

20 Polynomial Long Division SOLUTION: Insert the missing power with a zero coefficient Divide : 6x 4 - 3x 3 - x - 5 by 2x - 3 PROBLEM: Missing the x 2 term

21 Step 1: Write the problem using a division symbol Step 2: Look at the first term on the outside and the inside Polynomial Long Division Divide : 6x 4 - 3x 3 - x - 5 by 2x - 3

22 x times (what?) = 6x 4 Step 3:The outside term (x) was multiplied by (something) to equal (6x 4 ), the inside term. We must figure out what that (something) was. ? Put 3x 3 on the top So, the term we are looking for is 3x 3 Well, we started with one x and we ended up with x 4, so we picked up three more x’s or x 3. Also, the 2 changed into a 6, so we multiplied by 3. 3x 3

23 3x 3 6x 4 - 9x 3 Be sure to change the signs of every term. The next step is subtraction so we have: -( 6x 4 - 9x 3 ) = - 6x 4 + 9x 3 Multiply the term you just wrote on top by the outside terms. 3x 3 (2x - 3) = 6x 4 - 9x 3 - +

24 3x 3 + 3x 2 6x 3 + 0x 2 -6x 4 + 9x 3 Now we will repeat the whole process again. Step 1: look at the first terms Subtract (The first terms should always cancel 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

25 3x 3 + 3x 2 6x 3 + 0x 2 + 9x 2 - 6x 3 + 9x 2 Be sure to change the signs of every term. Bring down the 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 terms should always cancel out) -6x 4 + 9x 3 - x 17 2

26 3x 3 + 3x 6x 3 + 0x 2 + 9x 2 - 6x 3 + 9x 2 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 terms should always cancel out) -6x 4 + 9x 3 Bring down the next term If the coefficient of the outside term, 2x, does not go evenly into the coefficient of the inside term, 9x 2, then the number that goes on top will be: (inside/outside)= 9/2 + x x 2 + x x - x

27 ANSWER IS ON TOP 3x 3 + 3x 6x 3 + 0x 2 + 9x 2 - 6x 3 + 9x 2 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 terms should always cancel out) -6x 4 + 9x 3 + x x 2 + x x - x x No more terms to bring down, this (5) is the remainder + 5 2x-3 The remainder is written as a fraction. the remainder over the divisor (outside polynomial) REMAINDER DIVISOR

28 Practice Problems: (Hit enter to see the answers) Divide using Polynomial Long Division 1) 15x x - 5 by 3x + 5 2) 12x x - 35 by 2x - 7 3) 4x 3 - 2x - x by x - 2 4) 3x 3 - 5x x - 7 by 3x + 1 5) 5x 3 + 2x - 3 by x - 2 Answers: 1) 5x - 1 2) 6x + 5 3) 4x 2 + 7x ) x 2 - 2x - 7 5) 5x x

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