# Find a third-degree polynomial equation with rational coefficients that has the given numbers as roots. -5 and 1 - i

Braxton Pugh 2020-12-29 Answered
Find a third-degree polynomial equation with rational coefficients that has the given numbers as roots. -5 and 1 - i
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Raheem Donnelly

if1-i is a root, so its complex conjugate, $1+i$. Write factors $\left(x-k\right)$ where k is a root for all 3 roots. Then multiply together.
$\left(x-\left(-5\right)\right)\left(x-\left(1-i\right)\right)\left(x-\left(1+i\right)\right)=\left(x+5\right)\left({x}^{2}-x\left(1+i\right)-x\left(1-i\right)+1\left(1-i\right)\left(1+i\right)\right)$
$=\left(x+5\right)\left({x}^{2}-x-ix-x+ix+1i{i}^{2}\right)$
$=\left(x+5\right)\left({x}^{2}-2x+2\right)$
$={x}^{3}-2{x}^{2}+2x+5{x}^{2}-10x+10$
$={x}^{3}+3{x}^{2}-8x+10$