Help with finding out functions please.

Question

 Given that f(x)  is a cubic function with zeros at −6  and −2i+4  , find an equation for f(x) given that f(−7)=−6. 

 

I keep getting (x^3+6x^2+8x+48)/(9.5) but it is wrong.

 

Thanks in advance! 

Answer 1

One of the zeros is a conjugate of the other.  This conjugate is the third zero.  So your zeros are

 

x = -6

x = 4 - 2i

x = 4 + 2i

 

Think of the last two zeros being obtained using the quadratic formula.

 

Putting the function into factored form, we get

 

f(x) = C(x + 6)(x - [4 - 2i])(x - [4 + 2i])

 

where C is leading coefficient.  We have this value of C because of the initial value condition.  f(-7)=-6.

 

 

Expanding the function, we get

 

f(x) = C(x + 6)(x² - x(4 + 2i) - x(4 - 2i) + (16 + 4))

 

f(x) = C(x + 6)(x² - 4x - 2ix - 4x + 2ix + 20)

 

f(x) = C(x + 6)(x² - 8x + 20)

 

Notice that we no longer have imaginary terms.

 

 

f(x) = C(x³ - 8x² + 20x + 6x² - 48x + 120)

 

f(x) = C(x³ - 2x² - 28x + 120)

 

 

Next, solve for C using the initial condition.

 

-6 = C(-7 + 6)((-7)² - 8(-7) + 20)

 

-6 = C(-1)(49 + 56 + 20)

 

-6 = -125C

 

6 / 125 = C

 

 

Plugging in the value of C into the function,

 

f(x) = (6 / 125)(x³ - 2x² - 28x + 120)

 

 

Then distributing gives you the final form of the polynomial

 

f(x) = (6 / 125)x³ - (12 / 125)x² - (168 / 125)x + (144 / 25)

 

 

Answer 2

Are you saying there are 2 zeroes, -6 and (-2i+4)?

 

How did you get your answer?