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Complex Numbers

"Complex" numbers share many properties with the real numbers. In fact, complex numbers include all the real numbers, so you know lots of them already. Such an overachiever.

Factoring Polynomials

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 \( a + b\)?

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