Question

what is the sum of the roots of the polynomial shown below f(x)=3x^3+12x^2+3x-18

Answer 1

First you need to find the roots, or zeros, of the polynomial. In order to do that, you would want to factor the polynomial. Do you have an idea of how to factor this?

Answer 2

No idk how

Answer 3

Since all of the terms of this polynomial have a common factor of 3, you can begin by factoring out a 3:

3x^3 + 12x^2 + 3x - 18 = 3(x^3 + 4x^2 + x - 6)

Then you can look at this polynomial: x^3 + 4x^2 + x - 6 and use the rational roots theorem to create a list of possible roots. Are you familiar with the rational roots theorem?

Answer 4

no not really

Answer 5

Would it be written like this f(x)=x^3+4x^2+x-6 ?

Answer 6

What the rational roots theorem does is it allows you to narrow down your possible roots for a large polynomial. Here's how you create a list to test:

Look at the lead term (in this case x^3). The coefficient of the lead term is 1, since x^3 is the same as 1x^3. We will call this coefficient q, and we want to find factors of q. In this case, since q is 1, the only factor we have is 1.

Now we also look at the constant term, -6. We call this p. We want to find the factors of p. Factors of p are: -1,1,-2,2,-3,3,-6 and 6.

The list of possible roots is created by using these numbers and creating "ratios" between the numbers in set p and set q. So that gives us this list:

p/q: -1/1 or -1; 1/1 or 1; -2/1 or -2; 2/1 or 2; -3/1 or -3; 3/1 or 3; -6/1 or -6; and 6/1 or 6

We will test these by plugging them into the polynomial to see which one(s) give us 0. A "quick" method of doing this is called synthetic substitution.

Answer 7

In response to your question, yes that is how you would write your polynomial.

Answer 8

Sorry to be so long-winded. :) It is just this is a quick method once you get used to using it! And on larger polynomials, it is useful.

Answer 9

its fine :) im actually understanding it better

Answer 10

Now I am going to resort to a drawing to explain synthetic substitution. Please bear with me. :)

Answer 11

To test the numbers in your list, you create a structure that looks like this. Then bring down the first number (next diagram):