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Long Division © 2007 Thomas M. Kenyon Use with permission: [email protected] Review: How many times does 3 go into 24? 24 3 8 times How many times does 7 go into 35? 35 7 5 times How many times does -4 go into 12? 12 4 3 times How many times does 2x2 go into 8x5? 5 8x 3 8 x 2 x 2 4x times 2x 5 2 How many times does 3x go into 15x3? 15 x 3x 5 x times 3 2 Long division review: 68 5 74 50690 90 444 62 592 37 370 0 Steps: How many times does 74 go into 370? 506? 629? ans: 586 Multiply 5*74 6*74 8*74 ans: 370 444 592 Subtract Bring down the next DONE! Repeat the steps 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! A polynomial long division problem looks like this: (12 x 7 x 10) (3x 2) 2 or this: 3x 2 12 x 7 x 10 2 or this: 12 x 7 x 10 3x 2 2 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). 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. 3x 12 x 6 x 9 x 3 2 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. 4x2 -2x +3 3x 12 x 6x 6x 9x 3 2 12x3 0 -6x2 0 First, how many times 9x does 3x go into 12x3? We already 32 and divide it by 3x. saw atmany the beginning that3xwe take 12x How times does go into +9x? -6x ? 0 2 times. 2 by In this case, it goes into it 4x Divide Positive -6x 3 times. 3x to find out. In this case, -2x times. Second, we Multiply 3 times multiply 3x -2x 4x2 times 3x 3x. Third, we subtract Subtract subtract.again. Bring Now, we down bring thedown next term. the next Then term. repeat Repeat once the more. process. Done!!! Your turn: Answer: 3x2 -2x + 4 5x 15x 10x 10 x 20 20xx 3 22 15x3 0 -10x2 0 20x 0 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. 2x2 -4x +3 3 2 3x 5 6 x 2 x -11x 11x + 15 6x3 +10x2 0 (yikes!!) -12x2 -12x2 -20x 0 + 9x 3 How many times does 3x go into 6x ?? 9x + 15 2 by 3x Good Now, news!: multiply You 2x don’t have + 5.to worry You have about to multiply how by the Multiply You don’t need to worry about the 0 + 5+yet… Just, Subtract. 0 Subtract many entire times divisor, 3x+5 not goes just the into3x. something… That’s the You only only thinghave different Multiply… 3 Bring downtimes the next How many does term. 3x go into 6x ? to do “how many from times the previous does 3x two go9x? into. examples. . .” 2 How Notice: many we still times always does get 3x go a into Multiply -4x by the whole divisor Now, how many times does 3x go into -12x ? Be careful! When you subtract Bring down the next term Again, be careful with signs. Multiply thetimes 3 byterm the entire divisor… zero for the first when we 3about That is, -4x 3x + 5 Again, don’t worry the +of5 out. As usual, divide 6x by 3x to find aYou’re polynomial, you subtract all subtracting negative 20x. is thethe same! distributive property) 3Everything times 3x +else 5. (Use are subtracting. 2 divided -12x by 3x is -4x times. Subtract the terms. All the signs change. Subtracting changes the sign, so it’s the 2 2) (again, distributive property) This time, it’s 2x (the second term becomes – 10x same as adding 20x. property) (use the distributive Done! example 2: 5x2 -6x +7 2 x 4 10 x 32 x +38x 38x -28 28 3 2 10x3 -20x2 0 (yikes!!) -12x2 -12x2 + 24x 0 + 14x 14x - 28 + how 0 You don’t have to worry 0 about Multiply Good news!: Multiply… Subtract. Subtract many times 2x45x goes 2 byinto something… 3??You only have How Notice: Now, many multiply we still times always does get 2x go a 4. into You 10x have to multiply by the How many times does 2x go into 14x? 2? to Multiply do “how -6x many by the times whole does divisor 2x go into. . .” Now, how many times does 2x go into -12x Multiply the 7 by the entire divisor… zero entire for divisor, the first not term just when the 2x. we That’s the only thing different Bring down the next Be careful! Again, be careful When with youterm signs. subtract That is,down -6x-3times 2x -find 4 the Again, don’t worry about -of 4 7are times 2x 4. Divide subtracting. 10x from by the 2xnext previous to out. examples with monomials. Bring the term. a polynomial, You’re subtracting you subtract 24x. all 2 divided Subtract Everything else is2x the -12x bythe is same! -6xchange. times. the terms. All signs (again, distributive 2 property) (use the distributive property) This time, it’s 5x (Use the distributive (the second term becomes +20x2 ) property) Done! Your turn!! 3x2 -2x -4 +12 5x 3 15x 19 x -14x 14 x 12 3 2 15x3 -9x2 0 (yikes!!) -10x2 -10x2 + 6x 0 - 20x - 20x + 12 0 + 0