rational root theorem

Question

Use the rational root theorem to identify the possible roots. Verify with synthetic division and write as a product of its factors. Use the factors to find the remaining roots. Use additional paper for your work.

q(x)=2x^3+x^2-5x+2

Answer 1

The rational root theorem specifies that a polynomial that has integer coefficients should have roots related to the factors of the leading and trailing coefficients. If the polynomial has rational roots they should be a fraction made up of factors from the first and last ordered elements. That may be easier understood as the constant being multiplied by the highest power variable (in this case x^3) and the constant being multiplied by the lowest (x^0). For this problem both the highest and lowest power variables have coefficients of 2 (2x^3 and 2).

For this problem we would thus have the possibilities of 1, 2, and 1/2 and each could be positive or negative. Once we know that, we can divide to see what we get. At this point we have to essentially guess, but we have some more clues.

1. The first element is positive (2x^3) it's a positive 2
2. The last element is positive (2) it's also a positive 2
3. one of the elements in the middle is negative (-5x)

In order to get a positive number we need to multiply either all positives or an even number of negatives (assuming we're not using imaginary numbers, which is an assumption of the rational root theorem). Since we have a negative coefficient, that means we must have two negative factors, and only one positive one. So we can start guessing using a negative factor and see what happens when we divide.

Let's try x - 1:

x - 1 / 2x^3 + x ^2 -5x + 2 ==> multiply by 2x^2

2x^3 + x^2 -5x + 2

-2x^3 + 2x^2

==> 3x^2 -5x + 2 ==> multiply by 3x

3x^2 -5x +2

-3x^2 + 3x

==> -2x + 2 ==> multiply by -2

-2x + 2

'- -2x + 2

==> 0 we did it!

So now we pull down what we multiplied: (2x^2 + 3x - 2)(x-1) = 2x^3 + x^2 - 5x + 2

You may already know how to factor polynomials that are only to the 2nd power, but in case not, we can use similar logic to what we did before. We know the following:

1. We will have a positive and negative 0-power coefficient
2. one of the x-coefficients and 0-power coefficients will be a 2 (so we can get 2x^2)
3. The combined term will be a positive 3

At this point we could guess, but since we know we have (2x +/- ?)(x +/- ?) for the x-coefficients and (?x +/- 1)(?x +/- 2) for the 0-power and we know that we want positive 3 and we will have to subtract one product of one of the following (1 x 2), (2 x 2), or (1 x 1) from one of the others to get 4 and the products are 2, 4, and 1. The only answer we can use where we subtract one of the products and add the other to get 3 is 4 and -1. We must subtract one of them because we need a negative product, so we can't add 2 and 1, nor can we use -2 and -1 because that doesn't add up to 3, nor does it leave a negative 0-power coefficient. So we need 4 and 1 from our foil method and we need the 1 to be negative. That leaves us with one option:

(2x - 1)(x + 2).

The 2x * 2 gets us 4x and the x * -1 gets us -x. 4x - x = 3x. Altogether we get 2x^2 + 3x - 2, and that's what we wanted.

Now we take the first factor we used (x - 1) and include it with the two we just got ==> (x - 1)(2x - 1)(x + 2)

I know it's a bit long, and a bit late, but it's more because of the explanation than how long it takes to do. Hope that helps!