Given \( P(x)=x^{3}+5 x^{2}+9 x+45 \). Write \( P \) in factored form (as a product of linear factors). Be sure to write the full equation, including \( P(x)= \).
Solución de inteligencia artificial de Upstudy
Responder
Solución
Respondido por UpStudy AI y revisado por un tutor profesional


Extra Insights
To factor \( P(x)=x^{3}+5 x^{2}+9 x+45 \), we first look for rational roots using the Rational Root Theorem. Testing x = -3, we find that it is indeed a root. Thus, we perform polynomial long division of \( P(x) \) by \( (x + 3) \) and get \( P(x) = (x + 3)(x^2 + 2x + 15) \). Next, we find the roots of the quadratic \( x^2 + 2x + 15 \) using the quadratic formula, yielding complex roots \( -1 \pm 4i \). Therefore, the fully factored form of \( P(x) \) is: \[ P(x) = (x + 3)(x - (-1 + 4i))(x - (-1 - 4i)) \] So, the final factored form is: \[ P(x) = (x + 3)(x + 1 - 4i)(x + 1 + 4i) \]
