find a third-degree polynomial with real coefficients and with zeros -3 +i and -4 are zeros
Hello, yuppie!
Find a third-degree polynomial with real coefficients and with zeros $\displaystyle -3 +i$ and $\displaystyle -4.$
Complex roots always appear in conjugate pairs.
Hence, if $\displaystyle -3+i$ is a root, then $\displaystyle -3-i$ is also a root.
The cubic is: .$\displaystyle \big(x - [-4]\big)\,\big(x - [-3+i]\big)\,\big(x - [-3-i]\big)$
. . . . . . . . $\displaystyle =\;(x+4)(x+3-i)(x+3+i)$
. . . . . . . . $\displaystyle =\;(x+4)(x^2+6x+10)$
. . . . . . . . $\displaystyle =\;x^3 + 10x^2 + 34x + 40$