2.7: Consejos y trucos de álgebra Parte III (Factoring)
- Page ID
- 116846
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Al factorizar una expresión como esta:
\(x^2 - 8x + 15\)
El objetivo es escribir esto como\((x + a)(x + b)\) para algunos números\(a\) y\(b\), donde\(a\) y\(b\) podría ser positivo, negativo, o cero. Ya\((x + a)(x + b) = x^2 + (a + b)x + (ab)\) que vemos necesitamos\(a + b = -8\) y\(ab = 15\). De esa manera, cuando lo vuelves a flotar, tienes\(x^2 - 8x + 15\). Vemos si\(a = -3\) y\(b = -5\), esto funciona para ambos\(a + b = -8\) y\(ab = 15\). Por lo tanto,
\(x^2 - 8x + 15 = (x - 3)(x - 5)\)
Hagamos un par de ejemplos más.
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Factor\(x^2 + 3x + 2\).
En este caso queremos\(a + b = 3\) y\(ab = 2\). \(a = 1\)y\(b = 2\) funciona, entonces\(x^2 + 3x + 2 = (x + 1)(x + 2)\).
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Factor\(x^2 + 5x - 84\).
Esto es un poco más difícil porque los números son mayores, pero todavía podemos hacerlo. Queremos\(a + b = 5\), y\(ab = -84\). Podemos ver que\(84\) son\(12\) tiempos\(7\). Entonces si tenemos\(a = 12\) y\(b = -7\), entonces\(a + b = 5\) y\(ab = -84\). De ahí\(x^2 + 5x - 84 = (x + 12)(x - 7)\).
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Factor\(x^2 - 64\).
En este caso, queremos\(a + b = 0\) y\(ab = -64\). Pero fíjense que esto significa\(a = -b\), y por lo tanto\(-a^2 = -64\), lo que significa\(a^2 = 64\). Eso significa\(a = 8\), así\(b = -8\) (o viceversa). De ahí\(x^2 - 64 = (x + 8)(x - 8)\).