Two roots of the polynomial function f(x) = x3 − 7x − 6 are −2 and 3. use the fundamental theorem of algebra and the complex conjugate theorem to determine the number and nature of the remaining root(s). explain your thinking.

The degree of the polynomial is 3.

By the fundamental theorem of algebra, the function has three roots.

Two roots are given, so there must be one root remaining.

By the complex conjugate theorem, imaginary roots come in pairs.

The final root must be real.

Step-by-step explanation:

( x³ - 7 x - 6 ) : ( x + 2 ) = x² - 2 x - 3

-x³ - 2 x²

     - 2 x² - 7 x

        2 x² +4 x

                - 3 x - 6

                  3 x + 6

                  R(x) = 0

The polynomial: x³ - 7 x - 6  =  ( x + 2) ( x² - 2 x - 3  )

x² - 2 x - 3 = x² - 3 x + x - 3 = x ( x - 3 ) + ( x - 3 ) = ( x + 1 ) ( x - 3 )

The polynomial has roots : -2, 1, 3.

Step-by-step explanation:

Rewrite to help us solve.

( x - 5 )( 2x + 1 )( x - 7 ) = 0

Since these are all factored out it's easy to solve. We need to set each equal to 0 to find x.

x - 5 = 0                  2x + 1 = 0               x - 7 = 0

x = 5                        2x = - 1                   x = 7

                                x = -

So x - , 5 or 7

x also cannot be a negative number, but can be 0.