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Mastering Polynomial Long Division: Step-by-Step Guide

This presentation provides a comprehensive guide to performing polynomial long division. Controlled by the user, each step is revealed incrementally, allowing for a clear and focused understanding. The lesson covers various problems and techniques to ensure you grasp how to divide polynomials effectively. By following along, you'll learn how to set up the division, multiply terms, change signs, and subtract properly. This methodical approach will enhance your mathematical skills and confidence in handling polynomial expressions.

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Mastering Polynomial Long Division: Step-by-Step Guide

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  1. SLIDE SHOW INSTRUCTIONS • This presentation is completely under your control. • This lesson will show only one step at a time, • to see the next step you must press a key. • (Actual names written on a key are in green) • TO STOPTHE SLIDE SHOW: press ‘escape’ (Esc, top left of keyboard) • TO MOVE FORWARD: press the “spacebar” or Enter • (PageDn,  , , also work) • TO MOVE BACKWARD: press the  key • (PageUp, or also work)

  2. Polynomial Long Division

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

  4. ? Put 2x2 on the top 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 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

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

  6. Subtract (The first terms should always cancel out) -2x3 - 6x2 Bring down the next term 2x2 - x - 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

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

  8. Subtract (The first terms should always cancel out) Be sure to change the signs of every term. -2x3 - 6x2 + x2 + 3x - 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 - x - x2 + 2x + 6 0

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

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

  11. ? Put 3x4 on the top 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 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

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

  13. Subtract (The first terms should always cancel out) -3x5+ 18x4 Bring down the next term 3x4 + x3 - 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

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

  15. Subtract (The first terms should always cancel out) -3x5+ 18x4 Bring down the next term - x4 + 6x3 - 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 3x4 + x3 - 9x2 - 15x3 + x4 - 9x3 + 54x2 0x2 + 4x

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

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

  18. 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.

  19. Divide : 6x4 - 3x3 - x - 5 by 2x - 3 Polynomial Long Division 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!

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

  21. Divide : 6x4 - 3x3 - x - 5 by 2x - 3 Step 2: Look at the first term on the outside and the inside Polynomial Long Division Step 1: Write the problem using a division symbol

  22. ? Put 3x3 on the top 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 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

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

  24. Subtract (The first terms should always cancel out) -6x4 + 9x3 Bring down the next term 3x3 + 3x2 + 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

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

  26. 9 + x 2 Subtract (The first terms should always cancel out) -6x4 + 9x3 Bring down the next term Be sure to change the signs of every term. - 6x3 + 9x2 - x - 9x2+ x 27 17 2 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 3x3 + 3x 6x3 + 0x2 + 9x2 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

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

  28. Answers: 1) 5x - 1 2) 6x + 5 3) 4x2 + 7x + 12 + 4) x2 - 2x - 7 5) 5x2 + 10x + 22 + 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

  29. End of Tutorial Go to www.greenebox.com for more great math tutorials for your home computer Questions? send e-mail to: lgreene1@satx.rr.com

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