Question

Which polynomial function has a leading coefficient of 1 and roots (7 + i) and (5 – i) with multiplicity 1?

Answer 1

since (7 + i) and (5 - i) are complex roots.
we know, complex root is taken in conjugate.
so,their conjugate complex roots are (7- i) and (5 + i) respectively.

it means, we have four roots e.g., (7 + i) , (5 -i) , (7 - i) and (5 + i).
so, we have four factors,
x - (7 + i) , x - (5 - i) , x - (7 - i) and x + (5 + i)
now, multiply these four factors , as multiplicity is 1 , leaving 1 in front of product :

e.g., [x - (7 + i)][x - (5 - i)] [x - (7 -i)][x - (5 + i)]
= {x² - (7 + i + 7 - i)x + (7 + i)(7 - i)}{x² - (5 -i + 5 + i)x + (5 - i)(5 + i)}
= {x² - 14x + 50}{x² - 10x + 26}

Answer 2

The function will be f(x) = [x - (7 - i)] [x - (5 + i)] [x - (7 + i)] [x - (5 - i)]

The roots of the equation is 7 + i and 5 – i that are complex numbers.

Hence, the main roots that will come are,

X – (7 + i), x – (7 – i), x – (5 – i) and x – (5 + i)

All you need to do is to multiply the factors with each other.