List all possible rational roots for the given equation. x^4 - 2 x^2 - 16 x - 15 = 0

Question

ANSWER: _____?____

Use synthetic division twice to test the possible rational roots and find two actual roots.
The rational roots you found are x =____?___. (Separate your answers with a comma.)

Now use your quotient from the synthetic division to find the remaining zeros of the polynomial function.
The remaining zeros of f(x) are x =___?___ and x = ___?___

Answer 1

Set P is factors of 15. So P = {+-1, +-3, +-5, +-15}

Set Q is factors of 1.  So Q = {+-1}

 

P/Q = P = P = {+-1, +-3, +-5, +-15}

 

Now it's time to test to see which one of them make that polynomial zero.

For X=-1,  (-1)^4 - 2(-1)^2 - 16(-1) - 15 =  1 -2(1) + 16 - 15 = 1 - 2 + 16 - 15 = -1 + 16 - 15 = 15 - 15 = 0

 

So -1 is a rational root which means (x+1) is a factor.

 

Synthetic division by X+1 results in:

 

 -1                1    0     -2    -16    -15

                          -1     1       1    15

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

                    1    -1    -1    -15    0

 

 So the resulting cubic polynomial is X^3 - X^2 - X - 15 = 0

 

It turns out the same sets for P and Q apply to this cubic polynomial because they have the same

constant and leading coefficient.

This time it is X=3 that will make the polynomial zero, as 3^3 - 3^3 - 3-15 = 27 - 9 - 3- 15 = 18-3-15 = 15-15=0

 

Since 3 is the root of the cubic, X-3 is a factor

 

Synthetic division gives                            1    -1     -1     -15

                                                                       3     6       15

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

                                                               1     2      5       0

 

So the quadratic that results X^2 + 2x + 5

 

This cannot be factored, so the quadratic formula shows the last two roots are, with A=1, B=2, C=5:

 

   [-2 +/- square-root(2^2 - 4(1)(5)]/(2*1) =

 

[-2 +/- square-root(4 - 20]/2

 

[-2 +/- square-root(-16)]/2

 

[ -2 +/- 4i ]/2 =

 

 -1 +/ 2i

 

So the rational roots are -1, 3

 

The remaining imaginary roots are -1 +/- 2i

 

 

 

                  

Answer 2

To determine all possible rational roots of this polynomial, look at the factors of the integer coefficient of the leading term, which is 1 (the coefficient of x⁴), and the factors of the constant term, which is -15. If we represent each of the factors of the constant term by p and each of the factors of the leading coefficient by q, then the possible rational roots will be all possible quotients that can be formed by p/q.