Dividing one polynomial by anotherHow is this done? For example, how would one simplify the following? x 3 − 12 x 2 + 0 x − 42 x 2 − 2 x + 1 I can do it with long division, but it never makes intuitive sense to me. Either an explanation of the long division algorithm or a new way of solving this would be much appreciated.

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Dividing one polynomial by anotherHow is this done? For example, how would one simplify the following?                    x        3            −      12              x        2            +      0      x      −      42                      x        2            −      2      x      +      1      I can do it with long division, but it never makes intuitive sense to me. Either an explanation of the long division algorithm or a new way of solving this would be much appreciated.

Phoenix Morse · Accepted Answer

First step: By which monomial should you multiply       x    2    −  2  x  +  1 in order to get some polynomial with the same leading coefficient as       x    3    −  12      x    2    −  42? That&#039;s easy:   x. So, now you do      x    3    −  12      x    2    −  42  −  x  ×  (      x    2    −  2  x  +  1  )  =  −  10      x    2    −  x  −  42.Now, the same question: By which monomial should you multiply       x    2    −  2  x  +  1 in order to get some polynomial with the same leading coefficient as   −  10      x    2    −  x  −  42? Now, the answer is   −  10. So, now you do  −  10      x    2    −  x  −  42  −  (  −  10  )  ×  (      x    2    −  2  x  +  1  )  =  −  21  x  −  32.Now the degree is less than the degree of       x    2    −  2  x  +  1 and therefore there&#039;s nothing else to do: the quotient is   x  −  10 and the remainder is   −  21  x  −  32.

videosfapaturqz · Accepted Answer

x    3    −  12      x    2    −  42  =      x    3    −  2      x    2    +  x  −  10      x    2    +  20  x  −  10  −  21  x  −  32  =  =  (  x  −  10  )  (  x  −  1      )    2    −  21  x  −  32  ,which says                    x        3            −      12              x        2            −      42              (      x      −      1              )        2              =  x  −  10  +            −      21      x      −      32              (      x      −      1              )        2              .

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