A function f with real coefficients has the following characteristics

Question

A function f with real coefficients has the following characteristics

 

degree f= 4

f has a y-intercept of 52 

x-2 is a factor of f

-4/3 and 2-3i are both zeros of f 

 

Find f(x) and write in factored form as a product of 2 linear factors and an irreducible quadratic factor 

 

Answer 1

Degree 4 means you need 4 zeros.  Real coefficients means that if you have a complex zero, its complex conjugate must also be a zero.

 

So far,

 

f(x) = a(x - 2)(x + 4/3)([x - (2 - 3i)][x - (2 + 3i)],

 

with a real and a ≠ 0.

 

Note that

 

[x - (2 - 3i)][x - (2 + 3i)] = x² - (2 + 3i)x - (2 - 3i)x + (2² + 3²) = x² - 4x + 13

 

f(x) = a(x - 2)(x + 4/3)(x² - 4x + 13)

 

To find a, you need to know the value of y at some x.  I am not sure what you mean by "f has a y-integer of 52."  Could you check the original problem statement and see if it gives an (x,y) pair?