# How do I find the factors of a fourth degree polynomial?

How do I find the factors of a fourth degree polynomial? For example, -5x^4 - 20x^3 - 12x^2 + 8x + 5 = 0

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• Anonymous

Use Ruffini's method. It will work the same way, only that you'll get (if it is solvable) a third degree polynom which can be solved by Ruffini as well.

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• Hi,

You can start looking for roots or zeros by looking for possible rational roots. Possible rational roots are formed by using + or - factors of your constant divided by factors of your leading coefficient. In this case the only possible rational roots you get are +5, -5, +1, or -1. These are the only "nice" answers you can get. You can find which ones are actually roots by doing synthetic division. Zeros or roots will have a remainder of zero and the only one of these that works is -1, which is your first root. You could do this same thing by putting the equation into a graphing calculator and looking to see where it crosses the x axis, or where a point in the table has a Y value of 0. This still shows x = -1 is a root.

Unfortunately, there is no nice way to get the other roots without using your graphing calculator. However, using it, you can use the zero or root command, set left and right bounds to each side of a root, and allow the calculator to find each root for you one at a time. The roots to your particular equation were at x = -1, x = -3.089439099, x = -.525967045, and x = .6154061438. That means the factored form of your polynomial would be something like y = a(x + 1)(x + 3.089439099)(x + .525967045)(x - .6154061438). If you entered your original equation into your graphing calculator as Y1 and this last equation without the "a" as Y2, you could compare their y values at any x value. When x = 0, Y1 = 5 and Y2 = -1, or when x = -3, Y1 = 8 and Y2 = -1.6. It is easy to see their values differ by a constant value of -5, so a = -5.

The factored form of the equation would be:

y = -5(x + 1)(x + 3.089439099)(x + .525967045)(x - .6154061438)

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• Try to find the rational zeros by the theorem in the link below. Then use synthetic division to check if the possible roots in the theorem are zeros.

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• x = -1 will work, then you'll find (x+1)(-5x^3-15x^2+3x+5) and so on

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