polynomial function

Question

Find all the real and imaginary zeros for the polynomial function. 

f(x)=24x^3-32x^2-14x+12

The real and imaginary zeros for the polynomial function are __. 

(type an integer or a simplified fraction. use a comma to separate answers as needed.) 

 

 

help is greatly appreciated, thank you!

Answer 1

Set f(x) equal to zero.

 

24x³ - 32x² - 14x + 12 = 0

 

 

Factor f(x) by taking out the GCF.  GCF is 2.

 

2(12x³ - 16x² - 7x + 6) = 0

 

 

Set the term in parenthesis equal to zero.

 

12x³ - 16x² - 7x + 6 = 0

 

 

Using the rational root theorem, the possible roots are

 

c = ±1, ±2, ±3, ±6, ±1/2, ±3/2, ±3, ±1/3, ±2/3, ±1/4, ±3/4, ±1/6, ±1/12

 

 

where (x - c) is a factor of f(x).

 

Use synthetic division to divide the coefficients of f(x) by c.  The remainder must be zero in the division.

 

1/2 is a root, so we can factor f(x) as

 

(x - 1/2)(12x² - 10x - 12) = 0

 

 

Set the second factor equal to zero to find the other factors.

 

12x² - 10x - 12 = 0

 

2(6x² - 5x - 6) = 0

 

 

6x² - 5x - 6 = 0

 

 

Use the quadratic formula to solve for the root:

 

x = (-b ± √(b² - 4ac)) / 2a

 

where:

a = 6

b = -5

c = -6

 

Plug in these values into the formula to solve for the roots.  You will get 2 roots because of the plus/minus sign.  Since we use the quadratic formula, there is a possibility that these roots are complex.