Rational Zeros Question: Please help!

Question

List all possible rational zeros for the given function:

f(x) = 2x^3 + x^2 - 3x +1

Use synthetic division to test the possible rational zeros and find an actual zero

Then use your quotient from the synthetic division to find the remaining zeros of the polynomial function

Answer 1

Hi Emma,

 

Do you understand the Rational Root Theorem which allows us to list the possible rational zeroes for this function? If not you should try to find it in your text. 

 

That theorem allows us to identify the "possible" rational roots as {+1, -1, +1/2, -1/2}

 

Here is what the synthetic division looks like for testing 1/2:

 

1/2      2      1      -3      1

                   1       1      -1

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

           2       2       -2    0

 

Since the remainder is 0, then indeed 1/2 is a root. Synthetic division by the other list of possible roots reveals that in fact they are NOT roots.

 

The coefficients of the quotient appear on the last line of the synthetic division, i.e. 2, 2, -2.

 

That means the remaining factor of the original function is 2x² + 2x - 2.

 

Just to be clear, the original function could be written like this:

 

f(x) = (x - 1/2) (2x² + 2x - 2) 

 

To find the remaining two roots we have to solve: 2x² + 2x - 2 = 0 using the quadratic formula (or completing the square). Since every term is divisible by 2 we can solve x²+x-1 = 0.

 

That means in the quadratic formula a=1, b=1, c=-1.

 

The two remaining roots are irrational: (-1 + √5)/2 and (-1 - √5)/2.