Algebra

Complex Numbers

Complex Numbers - Factoring Polynomials

         

A quadratic polynomial with real coefficients has 19+8i 19 + 8 i as a root. If the other root has the form a+bi a + bi , where aa and bb are real numbers, what is the value of a+b a + b ?

If x(x49i)(xa+bi)=x38x2+97x, x (x-4-9i)(x-a+bi) = x^3 -8x^2+97x, what is a+b a + b ?

Details and assumptions

i i is the the imaginary unit, defined by i2=1 i^2 = -1 .

Suppose that 7x214x+182=c(xabi)(xa+bi),7x^2-14x+182=c(x-a-bi)(x-a+bi), where ii is the imaginary number that satisfies i2=1i^2=-1 and b>0. b > 0. What is the value of a+b+ca+b+c?

Suppose that x4+6x3+17x26x18=(x+a)(xa)(x+b+ci)(x+bci),\begin{aligned} & x^4+6x^3+17x^2-6x-18 \\ &= (x+a)(x-a)(x+b+ci)(x+b-ci), \end{aligned} where ii is the imaginary number that satisfies i2=1i^2=-1. What is the value of a+b+ca+b+c?

Details and assumptions

Assume a>0 a > 0 and c>0 c > 0 .

It is given that the cubic polynomial f(x) f(x) has real coefficients, and f(x)=0f(x) = 0 has a complex root 2815i 28 - 15 i . If the sum of all the complex, non-real roots of f(x)=0 f(x) = 0 can be written as a+bi a + bi , what is the value of a+b a + b?

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