Writing polynomial functions with complex zeros

Question

How do you write a polynomial function (f) of least degree that has rational coefficients, a leading coefficient of one, and the zeros 5, 5, and 4+i?

Answer 1

Since the polynomial has rational coefficients, complex roots occur in conjugate pairs, so, since 4 + i is a root, 4 - i must be a root as well.   

 

By the Factor Theorem, if c is a root of f(x), then x - c is a factor of f(x).  

 

Therefore, the polynomial with smallest possible degree and with leading coefficient 1 would be:

 

(x - 5)(x - 5)(x - (4 + i))(x - (4 - i))

 

= (x² - 10x + 25)((x - 4) - i)((x - 4) + i)

= (x² - 10x + 25)((x - 4)² - i²)

= (x² - 10x + 25)(x² - 8x + 16  + 1)

= (x² - 10x + 25)(x² - 8x + 17)

= X⁴ - 8X³ + 17X² -10X³ + 80X² - 170X + 25X² - 200X + 425

= X⁴ -18X³ +122X² - 370X + 425

Answer 2

Hi Shadic,

 

The zeros represent binomial factors of the polynomial function. 

 

Step 1: Set each "zero" in a binomial like this:  (x-5)(x-5)(x-(4+i)) and set it equal to zero.  Don't forget to include the zero 4-i, since it was stated that the polynomial has rational coefficients.  So, now you have (x-5)(x-5)(x-(4+i))(x-(4-i)).

 

Step 2: (x-5)(x-5)(x-(4+i))(x-(4-i))=0, so after distributing (x-5)(x-5)((x-4)-i)((x-4)+i)=0

 

Step 3: (x-5)(x-5)[(x-4)²-(i)²]=0, so after multiplying you have (x²-10x+25)(x²-8x+16+1)=0

 

Step 4: Before combining like terms: x⁴-8x³+17x²-10x³+80x²-170x+25x²-200x+425=0

 

Step 5: After combining like terms: x⁴-18x³+122x²-370x+425=0

 

Does that make sense to you?