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Polynomial Long Division

Polynomial Long Division Divide : 2 x 3 + 5 x 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

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

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

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

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

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 2 x 2 - x + 2 -2 x 3 - 6 x 2 - x 2 + 3 x + x Subtract (The first terms + 2 x + 6 should always cancel out) - 2 x - 6 0 Be sure to change the signs of every term.

Polynomial Long Division Divide : 3 x 5 - 17 x 4 - 15 x 3 + 4 x + 54 x 2 - 24 by x - 6 PROBLEM: Terms out of descending order SOLUTION: Rearrange terms into descending order

Polynomial Long Division Divide : 3 x 5 - 17 x 4 - 15 x 3 + 54 x 2 + 4 x - 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

Step 3: The outside term (x) was multiplied by (something) to equal (3 x 5), the inside term. We must figure out what that (something) was. x times (what? ) = 3 x 5 3 x ? 4 Put 3 x 4 on the top 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. So, the term we are looking for is 3 x 4

Multiply the term you just wrote on top by the outside terms. 3 x 4(x - 6) = 3 x 5 - 18 x 4 3 x 4 - 3 x 5 +- 18 x 4 The next step is subtraction so we have: -(3 x 5 - 18 x 4) = -3 x 5+ 18 x 4 Be sure to change the signs of every term.

Subtract (The first terms should always cancel out) 3 x 4 + x 3 -3 x 5+ 18 x 4 Bring down + x 4 - 15 x 3 the next term Now we will repeat the whole process again. Step 1: look at the first terms Step 2: what did we multiply the outside term by to get the inside term. Step 3: Write this term on top

3 x 4 + x 3 5+ 18 x 4 Bring down -3 x Subtract sure to + x 4 - 15 x 3 the next. Beterm (The first terms should always change the 4 3 - x + 6 x cancel out) signs of Step 4: Multiply this new - 9 x 3 + 54 x 2 every term by the outside terms Step 5: Change the signs & write the answer under the current inside term

Repeat Steps 1 -5 Step 1: Look at first terms 3 x 4 + x 3 - 9 x 2 5+ 18 x 4 -3 x Bring down Step 2: What did we multiply by? + x 4 - 15 x 3 the next term Step 3: Write this 4 + 6 x 3 x Subtract above the line (The first terms 3 + 54 x 2 9 x Step 4: Multiply should always cancel out) - 9 x 3 - 54 x 2 new term by outside terms 2 + 4 x 0 x Step 5: Change signs & subtract 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 3 x 4 + x 3 - 9 x 2 + 0 x -3 x 5+ 18 x 4 Bring down + x 4 - 15 x 3 the next term - x 4 + 6 x 3 - 9 x 3 + 54 x 2 - 9 x 3 - 54 x 2 Subtract 2 + 4 x (The first terms + 0 x always Be should sure to 2 + 0 x cancel out) 0 x change the + 4 x - 24 signs of every term.

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 3 x 4 + x 3 - 9 x 2 + 0 x + 4 -3 x 5+ 18 x 4 + x 4 - 15 x 3 - x 4 + 6 x 3 - 9 x 3 + 54 x 2 - 9 x 3 - 54 x 2 + 0 x 2 + 4 x Subtract- 0 x 2 + 0 x (The first terms + 4 x - 24 should Be sure to always cancel out) - 4 x + 24 change the signs of every term. 0

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.

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

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

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

Step 3: The outside term (x) was multiplied by (something) to equal (6 x 4), the inside term. We must figure out what that (something) was. x times (what? ) = 6 x 4 ? 3 x 3 Put 3 x 3 on the top 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. So, the term we are looking for is 3 x 3

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

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

3 x 3 + 3 x 2 4 + 9 x 3 Bring down -6 x Subtract 3 Be term sure to 2 the next (The first terms 6 x + 0 x should always change the 3 2 cancel out) - 6 x + 9 x signs of 17 Step 4: Multiply this new + 9 x 2 x every term. 2 term by the outside terms Step 5: Change the signs & write the answer under the current inside 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 3 x 3 + 3 x + 9 2 x Bring down -6 x 4 + 9 x 3 the next term 6 x 3 + 0 x 2 3 + 9 x 2 6 x Subtract Be sure to 17 (The first terms 2+ 9 x x change the should always 2 cancel out) signs of 27 2 - 9 x + 2 x every term. 5 x - 5 If the coefficient of the outside term, 2 x, does not go evenly into the coefficient of the inside term, 9 x 2, then the number that goes on top will be: (inside/outside)= 9/2

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 9 5 3 3 x + 2 x-3 DIVISOR -6 x 4 + 9 x 3 Be sure to 3 2 6 x + 0 x change the - 6 x 3 + 9 x 2 sign of 17 2 + 9 x - 2 x every term. 2+ 27 x 9 x Subtract 2 (The first terms 5 x - 5 should always cancel out) - 5 x + 15 2 No more terms REMAINDER to bring down, this (5) is the remainder 5 The remainder is written as a fraction. the remainder over the divisor (outside polynomial)

Practice Problems: (Hit enter to see the answers) Divide using Polynomial Long Division 1) 15 x 2 + 22 x - 5 by 3 x + 5 2) 12 x 2 - 32 x - 35 by 2 x - 7 3) 4 x 3 - 2 x - x 2 + 6 by x - 2 4) 3 x 3 - 5 x 2 - 23 x - 7 by 3 x + 1 5) 5 x 3 + 2 x - 3 by x - 2 Answers: 1) 5 x - 1 2) 6 x + 5 3) 4 x 2 + 7 x + 12 + 4) x 2 - 2 x - 7 5) 5 x 2 + 10 x + 22 +

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