Long Division This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714 © 2007 Thomas M. Kenyon Use with permission: tkenyon@crcs.wnyric.org

How many times does 3 go into 24? Review: How many times does 3 go into 24? How many times does 7 go into 35? How many times does -4 go into 12? How many times does 2x2 go into 8x5? This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714 How many times does 3x go into 15x3?

6 8 5 74 50690 9 444 62 592 37 370 Long division review: Steps: DONE! 74 50690 9 How many times does 74 go into 370? ans: 5 How many times does 74 go into 506? ans: 6 How many times does 74 go into 629? ans: 8 444 62 Multiply 5*74 ans: 370 Multiply 8*74 ans: 592 Multiply 6*74 ans: 444 592 37 Subtract Subtract Subtract 370 This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714 Bring down the next Bring down the next DONE! Repeat the steps Repeat the steps

74 50690 Wouldn’t it be easier if we only had to worry comment: 74 50690 Wouldn’t it be easier if we only had to worry about the 7? The 4 makes it harder… It’d be a lot easier if we just had to worry about how many times 70 goes into 506 (7*70=490) Oh well, we weren’t that lucky in 5th grade, but we’re going to be that lucky when we do this with polynomials! This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714

A polynomial long division problem looks like this: or this: or this: This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714 All three notations mean the same thing: division. Note: we are not dividing by a monomial this time! We already learned division by a monomial. This time, the divisor is a polynomial (we’re going to stick to binomials).

However, we’re going to learn a new way to divide comment: However, we’re going to learn a new way to divide by a monomial. We will adapt this new method to learning long division by a polynomial with 2 or more terms in it. The steps for long division of polynomials are the same as the steps for long division of numbers: First, how many times does it go into the front part? Second, multiply Third, subtract Fourth, bring down the next term (number) Repeat these four steps until you’re done. This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714

4x2 -2x + 3 12x3 -6x2 9x First, how many times does 3x go into 12x3? We already saw at the beginning that we take 12x3 and divide it by 3x. In this case, it goes into it 4x2 times. How many times does 3x go into +9x? Positive 3 times. How many times does 3x go into -6x2? Divide -6x2 by 3x to find out. In this case, -2x times. This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714 Multiply 3 times 3x Second, we multiply 4x2 times 3x. Second, we multiply -2x times 3x Subtract Third, we subtract. Third, we subtract again. Bring down the next term. Then repeat once more. Now, we bring down the next term. Repeat the process. Done!!!

3x2 -2x + 4 15x3 -10x2 20x Your turn: Answer: -10x2 This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714 20x Note: in both of these problems, when we subtracted, we got zero every time! That’s why long division with polynomials is easier than long division with numbers.

Done! 2x2 -4x + 3 -11x + 15 6x3 +10x2 0 -12x2 -12x2 -20x 0 + 9x (yikes!!) -12x2 -20x 0 + 9x How many times does 3x go into 6x3?? You don’t need to worry about the + 5 yet… Just, How many times does 3x go into 6x3? As usual, divide 6x3 by 3x to find out. This time, it’s 2x2 9x + 15 Good news!: You don’t have to worry about how many times 3x+5 goes into something… You only have to do “how many times does 3x go into. . .” Everything else is the same! Now, multiply 2x2 by 3x + 5. You have to multiply by the entire divisor, not just the 3x. That’s the only thing different from the previous two examples. (Use the distributive property) Multiply Multiply -4x by the whole divisor That is, -4x times 3x + 5 (again, distributive property) 0 + 0 Subtract. Be careful! When you subtract a polynomial, you subtract all of the terms. All the signs change. (the second term becomes – 10x2 ) Subtract Again, be careful with signs. You’re subtracting negative 20x. Subtracting changes the sign, so it’s the same as adding 20x. Multiply… Multiply the 3 by the entire divisor… 3 times 3x + 5. (use the distributive property) Bring down the next term. This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714 Notice: we still always get a zero for the first term when we are subtracting. How many times does 3x go into 9x? Now, how many times does 3x go into -12x2? Again, don’t worry about the + 5 -12x2 divided by 3x is -4x times. Bring down the next term Done! Subtract

Done! +38x -28 5x2 -6x + 7 10x3 -20x2 0 -12x2 -12x2 + 24x 0 + 14x example 2: 5x2 -6x + 7 +38x -28 10x3 -20x2 0 -12x2 (yikes!!) -12x2 + 24x 0 + 14x 14x - 28 0 + 0 Multiply Multiply -6x by the whole divisor That is, -6x times 2x - 4 (again, distributive property) Good news!: You don’t have to worry about how many times 2x- 4 goes into something… You only have to do “how many times does 2x go into. . .” Everything else is the same! Multiply… Multiply the 7 by the entire divisor… 7 times 2x - 4. (use the distributive property) Subtract. Be careful! When you subtract a polynomial, you subtract all of the terms. All the signs change. (the second term becomes +20x2 ) Subtract Again, be careful with signs. You’re subtracting 24x. How many times does 2x go into 10x3?? Divide 10x3 by 2x to find out. This time, it’s 5x2 Notice: we still always get a zero for the first term when we are subtracting. Now, multiply 5x2 by 2x - 4. You have to multiply by the entire divisor, not just the 2x. That’s the only thing different from the previous examples with monomials. (Use the distributive property) How many times does 2x go into 14x? Now, how many times does 2x go into -12x2? Again, don’t worry about the - 4 -12x2 divided by 2x is -6x times. Bring down the next term Done! Bring down the next term. Subtract

-14x 3x2 -2x - 4 +12 15x3 -9x2 0 -10x2 -10x2 + 6x 0 - 20x - 20x + 12 Your turn!! 3x2 -2x - 4 -14x +12 15x3 -9x2 0 -10x2 (yikes!!) -10x2 + 6x 0 - 20x - 20x + 12 0 + 0

This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714 This presentation may be used by k-12 public education teachers. Contact me for permission to use this presentation in whole or in part for any other purpose. Feel free to send suggestions or to comment on any errors/omissions. tkenyon@crcs.wnyric.org or tomkenyon@gmail.com Thomas M. Kenyon 6455 Habgood Road Black Creek, NY 14714