SOLUTION: Hello! A polynomial P is given: P(x) = 2x^3 + 7x^2 + 4x - 4 Find all the real zeros of P thanks for the homework help!

Click here to see ALL problems on Polynomials-and-rational-expressions

Question 199776: Hello!
A polynomial P is given:
P(x) = 2x^3 + 7x^2 + 4x - 4
Find all the real zeros of P
thanks for the homework help!
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
First, let's find the possible rational roots.

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 -4 (the last coefficient):

Now let's list the factors of 2 (the first coefficient):

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

Now simplify

These are all the distinct rational zeros of the function that could occur

-----------------------------------------------------------------------

Now let's test the possible roots to see if they are actually roots.

Let's see if the possible zero is really a root for the function

So let's make the synthetic division table for the function given the possible zero :

1 | 2 7 4 -4
| 2 9 13
2 9 13 9

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of

------------------------------------------------------

Let's see if the possible zero is really a root for the function

So let's make the synthetic division table for the function given the possible zero :

1/2 | 2 7 4 -4
| 1 4 4
2 8 8 0

Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of

So this means that

Note: the quotient of the division results from taking half of the first three values in the bottom row.

Note: if you didn't find a root, then you would have to keep going until either you find one or you are done with the list.

Now let's solve to find the next two zeros:

Start with the given equation.

Notice we have a quadratic in the form of where , , and

Let's use the quadratic formula to solve for "x":

Start with the quadratic formula

Plug in , , and

Square to get .

Multiply to get

Subtract from to get

Multiply and to get .

Take the square root of to get .

or Break up the expression.

or Combine like terms.

or Simplify.

So the two solutions are or

We can simply write this as with a multiplicity of 2

===========================================================================

Answer:

So the three zeros of are: and (with a multiplicity of 2)

If you have any questions, email me at jim_thompson5910@hotmail.com