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Algebra

# Complex Numbers - 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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