how do you find possible linear factors for 5x cubed-12x squared-23x+42

Question

Trying to find the possible factors for the above listed problem:

5xcubed-12x squared-23x+42

Answer 1

You look for rational zeroes. You only need to find one to solve the problem.

Take the factors of the leading power"s coefficient: 5 has factors 1 and 5.

Take the factors of the lowest power's coefficient (the constant): 42 has factors 1, 42, 2, 21, 3, 14, 6, 7.

The rational zeroes are 1, 42, 2, 21, 3, 14, 6, 7, 1/5, 42/5, 2/5, 21/5, 3/5, 14/5, 6/5, 7/5 in positive and negative form. Substitute each one for x until you get zero in the function.

x=1 gives us: 5-12-23+42=12 (not zero): x=-1 gives us: -5-12+23+42=48 (not zero).

x=2 gives us 40-48-46+42=-12; x=-2 gives us -40-48+46+42=0 (bingo!). x=-2 is a zero. Divide by this zero using synthetic division:

-2 | 5 -12 -23    42

      5 -10  44 | -42

      5 -22  21 |   0 = 5x²-22x+21 = (5x-7)(x-3).

So we have all the zeroes: -2, 7/5, 3.