Complex Numbers
A quadratic polynomial with real coefficients has \( 19 + 8 i \) as a root. If the other root has the form \( a + bi \), where \(a\) and \(b\) are real numbers, what is the value of \( a + b \)?
If \[ x (x-4-9i)(x-a+bi) = x^3 -8x^2+97x, \] what is \( a + b \) ?
Details and assumptions
\( i \) is the the imaginary unit, defined by \( i^2 = -1 \).
Suppose that \[7x^2-14x+182=c(x-a-bi)(x-a+bi),\] where \(i\) is the imaginary number that satisfies \(i^2=-1\) and \( b > 0. \) What is the value of \(a+b+c\)?
Suppose that \[\begin{align} & x^4+6x^3+17x^2-6x-18 \\ &= (x+a)(x-a)(x+b+ci)(x+b-ci), \end{align} \] where \(i\) is the imaginary number that satisfies \(i^2=-1\). What is the value of \(a+b+c\)?
Details and assumptions
Assume \( a > 0 \) and \( c > 0 \).
It is given that the cubic polynomial \( f(x) \) has real coefficients, and \(f(x) = 0 \) has a complex root \( 28 - 15 i \). If the sum of all the complex, non-real roots of \( f(x) = 0 \) can be written as \( a + bi \), what is the value of \( a + b\)?