1)) Factor \( x^{4}+12 x^{2}+27 \) completely. All factors in your answer should have integer coefficients.
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
The Deep Dive
To factor \( x^{4}+12 x^{2}+27 \), we can start by making a substitution to simplify the expression. Let \( y = x^{2} \). The expression then becomes \( y^{2} + 12y + 27 \). We will look for two numbers that multiply to \( 27 \) (the constant term) and add up to \( 12 \) (the coefficient of the linear term). The numbers \( 3 \) and \( 9 \) satisfy both conditions since \( 3 \cdot 9 = 27 \) and \( 3 + 9 = 12 \). Now we can factor the quadratic expression: \[ y^{2} + 12y + 27 = (y + 3)(y + 9) \] Substituting back \( y = x^{2} \), we get: \[ (x^{2} + 3)(x^{2} + 9) \] Thus, the completely factored form of the original expression is: \[ \boxed{(x^{2} + 3)(x^{2} + 9)} \]
