How do you write a polynomial in standard form given the zeros 1 and 4-4i?

Mar 10, 2017

${x}^{3} - 9 {x}^{2} + 40 x - 32$

Explanation:

Complex zeros always come in pairs: $4 \pm 4 i$

Always multiply the complex zeros first to get rid of the complex terms by remembering that ${i}^{2} = - 1$

Multiply the complex factors: $\left(x - 4 - 4 i\right) \left(x - 4 + 4 i\right)$

$= {x}^{2} - 4 x + 4 i x - 4 x + 16 - 16 i - 4 i x + 16 i - 16 {i}^{2}$

$= {x}^{2} - 8 x + 16 - 16 \left(- 1\right)$
$= {x}^{2} - 8 x + 32$

Add the $\left(x - 1\right)$, $\left(x = 1\right)$ factor to the equation:

$\left(x - 1\right) \left({x}^{2} - 8 x + 32\right)$

Multiply using distribution:
${x}^{3} - 8 {x}^{2} + 32 x - {x}^{2} + 8 x - 32$

${x}^{3} - 9 {x}^{2} + 40 x - 32$