Questions: Factoriza el siguiente trinomio completamente 165-5 x^2-40 x

Factoriza el siguiente trinomio completamente 165-5 x^2-40 x
Transcript text: Factoriza el siguiente trinomio completamente \[ 165-5 x^{2}-40 x \]
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Solution

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To factor the given trinomial completely, we first look for a common factor in all the terms. Then, we factor the quadratic expression by finding two numbers that multiply to the product of the coefficient of \(x^2\) and the constant term, and add to the coefficient of the linear term.

Paso 1: Identificación del trinomio

Dado el trinomio \(165 - 5x^2 - 40x\), primero reorganizamos los términos para facilitar la factorización. Esto se convierte en \(-5x^2 - 40x + 165\).

Paso 2: Factor común

Observamos que todos los términos tienen un factor común de \(-5\). Sacamos este factor: \[ -5(x^2 + 8x - 33) \]

Paso 3: Factorización del trinomio cuadrático

Ahora, factorizamos el trinomio cuadrático \(x^2 + 8x - 33\). Buscamos dos números que multiplicados den \(-33\) y sumados den \(8\). Estos números son \(11\) y \(-3\). Por lo tanto, podemos escribir: \[ x^2 + 8x - 33 = (x - 3)(x + 11) \]

Paso 4: Expresión final

Sustituyendo de nuevo en la expresión con el factor común, obtenemos: \[ -5(x - 3)(x + 11) \]

Respuesta Final

La factorización completa del trinomio es: \[ \boxed{-5(x - 3)(x + 11)} \]

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