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.

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.

The answer is in the image 

SI=(P*R*T)/100

P=2000

R=1.5

T=6

SI=(2000*1.5*6)/100

=(2000*9)/100

=180

Neil will earn interest of 180

The answer is in the image 

The answer is in the image 

The answer is in the image 

For every 8 cars there are 7 trucks

Therefore,

Cars:Truck=8:7

Answer is B)8:7

The solution is in the following image