solve problem

Question

The polynomial function f(x) has the given zero, find the other zero?

 

f(x)=x³+8x²-3x-24; -8

Answer 1

To find the roots, we set f(x) equal to zero.

 

x³  + 8x² - 3x - 24 = 0

 

 

Other than using division methods, we can also find the other zeroes by group factoring.

 

x²(x + 8) - 3(x + 8) = 0

 

 

Notice that (x + 8) is a common factor of the equation.  If we set this factor equal to zero, we verify that -8 is a root.

 

 

Factor the equation using the common factor.

 

(x + 8)(x² - 3) = 0

 

 

Set the factors equal to zero.

 

x + 8 = 0            and             x² - 3 = 0

 

x = -8                 and             x² = 3

 

                                            x = -√(3)

                                            x = √(3)

 

The other solutions are

 

x = -√(3)                                           

x = √(3)

Answer 2

Since -8 is a zero of f(x), x-(-8) = x + 8 is a factor of f(x).

 

Divide f(x) synthetically by x + 8:

 

-8⌋    1      8      -3       -24

 

       ____     -8___ 0____24_____

         1      0       -3          0

 

So, f(x) = (x + 8)(x² - 3)

 

To find the remaining zeroes of f(x), solve the equation x² - 3 = 0.

 

If x² - 3 = 0, then x² = 3.     So, x = ±√3

Answer 3

Using synthetic substitution, the suppressed polynomial is:

 

x² - 3

 

{±√3}