How do you factor and find the zeros for the polynomial 2x^5-18x^4+14x^3+186x^2-160x-600?

Question

Please help with this and show all the steps. Thank you.

Answer 1

When factoring a polynomial this large, you'll probably want to use synthetic division. First off, however, you should always factor out any common factors to all terms. In this case, you can factor out a 2, then use synthetic division on the rest.

 

2(x⁵ - 9x⁴ + 7x³ + 93x² - 80x - 300)

 

Your synthetic division setup will look something like:

 

   |  1    -9    7    93    -80    -300

                                                  

                                          |

 

Remember the process of synthetic division is to add a column down, then multiply it's total by the chosen factor, and write that into the next column.

 

The possible factors you can try to divide out are factors of -300 (yuck) divided by factors of 1 (easy!).

x=±1,2,3,4,5,6,8,10,12,15,20...,300

Hopefully you won't even need that many factors. Because you need the final result to be 300, it's not likely that multiplying by 1 will get you there fast enough. Let's try 2

 

 2 | 1 -9     7    93   -80  -300
          2  -14    14   214   268
      1 -7     -7  107  134 | -32

 

The remainder isn't 0, so you have to keep trying. Lets try -2 next.

 

-2 | 1  -9      7    93    -80 -300
          -2     22  -58    -70  300
      1  -11   29    35  -150 |  0

 

Excellent, the remainder is 0! One of the zeros is x = -2, so a factor of (x+2)

 

You can now continue this with the dividend of the previous synthetic division:

 

___|  1   -11   29   35  -150

_______________________

                                |

 

x=±1,2,3,4,5,6,8,10,12,15,20...,150

Your list of possible zeros will shrink each time you successfully find a correct zero. Repeat this until you have a quadratic (3 terms), and you can factor in the usual way.