How do you factor polynomials?

Question

I had a little problem with polynomials. For example, how do you solve 2x^3 - 3x^2 - 5x = 0? My textbook tells me to guess the first solution for (x-r1)(x-r2)(x-r3), which is the factored form of the polynomial. But when you can't think of anything, what do you do?

Answer 1

There are many ways. Later on it won't matter what method to use. The form in your textbook is correct, it is a model. The r's are different numbers.

2X^3 -3X^2 -5X = 0

(X-0)(2X-5)(X+1) = 0  

X(2X-5)(X+1) = 0

Answer 2

First thing you want to do is look for common factors.  Is there anything you can divide out of every term?  If so, do it.  Let's see; every term on the left can be divided by x.

2x^3 - 3x^2 - 5x = 0

x(2x^2 - 3x -5) = 0

Next thing to do is look inside the brackets and product & sum factoring.  Can you think of two numbers whose product is (2)x(-5)=-10 and which have a sum of -3?  I think 2 and -5 fit the bill here.  So now we split up the middle (x) term using these two values.

x(2x^2 + 2x - 5x - 5)=0

Now factor by grouping on the first two terms and then the last two terms.  Look for common factors in each group.  Note that the sign on the x factor of the second group is the sign in the factoring.

x(2x(x+1) - 5(x+1))=0

If things are correct so far, the value in the brackets should be the same.  Yup, both are x+1.  Keep going.

Common factor the two inter terms and bring together what is left.

x(x+1)(2x-5)=0

So now what values of x make each factor 0, as the only way to get the whole thing to be zero is if one of the factors goes to 0.

So,

x = 0 or x+1 =0 or 2x-5 =0

x = 0 or x = -1 or x = 5/2

Hope that helps.

Answer 3

a simple method, especially if you make the right side zero, is to graph the function in a graphing calculator and finding the zero points on it.

for example if the function crossed the x axis at 1 then you would see one of the roots is (x-1).