First find all possible rational roots. To do this, find all the factors of the lowest order coefficient and the highest order coefficient. For #1, the highest order coefficient is 1 because the x^3 doesn't have a number in front of it. The lowest order coefficient is 30.

Here are all the factors:

Factors of 1 are: 1

Factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30

Now divide each factor of 30 (positive and negative), and divide them by each factor of 1.

All possible rational roots are:

-1, 1, -2, 2, -3, 3, -5, 5, -5, 6, -10, 10, -15, 15, -30, 30

Now we perform synthetic division like you have started to do. Try dividing the polynomial by each possible root. If the result has a remainder, that possible root does NOT work. Try another possible root. If there is not a remainder, you have found one of the roots.

For example, when dividing x^3 - 4x^2 -11x + 30 by the possible root 2, we get x^2 - 2x - 15 without a remainder. That means 2 is a root. From here we can factor the result to (x-5)(x+3).

So the roots for #1 are x = -3, 2, and 5.

Let me know if you need help with the others :)

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3 years ago