SOLUTION: Find all the zeros of the function: *x^3+3x^2-x+12

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Question 90672: Find all the zeros of the function:
*x^3+3x^2-x+12
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients

So let's list the factors of 12:

Now let's list the factors of 1:

Now let's divide each factor of the last coefficient by each factor of the first coefficient

Now simplify

These are all the possible zeros of the function

To save time, I'm only going to use synthetic division on the possible zeros that are actually zeros of the function.
Otherwise, I would have to use synthetic division on every possible root (there are 12 possible roots, so that means there would be at most 12 synthetic division tables).
However, you might be required to follow this procedure, so this is why I'm showing you how to set up a problem like this

If you're not required to follow this procedure, simply use a graphing calculator to find the roots

So with a graphing calculator, we find a root x=-4. So our test zero is -4

Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.

-4 | 1 3 -1 12
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)

-4 | 1 3 -1 12
|
1

Multiply -4 by 1 and place the product (which is -4) right underneath the second coefficient (which is 3)

-4 | 1 3 -1 12
| -4
1

Add -4 and 3 to get -1. Place the sum right underneath -4.

-4 | 1 3 -1 12
| -4
1 -1

Multiply -4 by -1 and place the product (which is 4) right underneath the third coefficient (which is -1)

-4 | 1 3 -1 12
| -4 4
1 -1

Add 4 and -1 to get 3. Place the sum right underneath 4.

-4 | 1 3 -1 12
| -4 4
1 -1 3

Multiply -4 by 3 and place the product (which is -12) right underneath the fourth coefficient (which is 12)

-4 | 1 3 -1 12
| -4 4 -12
1 -1 3

Add -12 and 12 to get 0. Place the sum right underneath -12.

-4 | 1 3 -1 12
| -4 4 -12
1 -1 3 0

Since the last column adds to zero, we have a remainder of zero. This means is a factor of

Now lets look at the bottom row of coefficients:

The first 3 coefficients (1,-1,3) form the quotient

So

You can use this online polynomial division calculator to check your work

Now let's use the quadratic formula to solve

Starting with the general quadratic

the general solution using the quadratic equation is:

So lets solve ( notice , , and )

Plug in a=1, b=-1, and c=3

Negate -1 to get 1

Square -1 to get 1 (note: remember when you square -1, you must square the negative as well. This is because .)

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root (note: since we cannot take the square root of a negative value, we must factor to to make the radicand positive. If you need help with simplifying the square root, check out this solver)

Multiply 2 and 1 to get 2

After simplifying, the quadratic has roots of

or

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So the polynomial has the roots

, or