Complex numbers help ??

Irish Fox

Need help with these questions

f(z) = z^2 + 6z + 25 is a quadratic polynomial

verify that z = - 3 - 4i is a root of f(z)

Write down the other root of the equation

z^2 + 6z + 25 = 0

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f(z) = z^3 - z^2 - 4z - 6 is a cubic polynomial

show that z = 1 + i is a solution to the equation f(z) = 0

MathsManiac

To verify that a number is a root of a quadratic or cubic equation, you substitute it in for the variable and show that it works.

So, for the first one, you need to show, by working it out, that
(-3-4i)^2 + 6(-3-4i) +25 = 0.

For the follow-on bit, you use a theorem called the "conjugate roots theorem". This says that in quadratic or cubic where the coefficients are real numbers, then the conjugate of a root is also a root. That is, if one root is a+bi, then a-bi is also a root.

Your second question is similar to the first: substitute and work it out to verify.