Question

1. what is the gcf of $x^4 + 5x^3 + 6x^2$?

1. which expression is equivalent to $x^4 - 6x^3 + 9x^2$?

a. $x^2(x + 3)^2$
b. $x^2(x - 3)^2$
c. $x^2(x + 3)(x - 3)$
d. $x^2(x^2 + 3)(x - 3)$

Explanation:

Question 4:

Step 1: Analizar términos

Los términos son \(x^4\), \(5x^3\) y \(6x^2\).

Step 2: Encontrar GCF de coeficientes y variables

• Coeficientes: 1, 5, 6. GCF(1,5,6) = 1.
• Variables: \(x^4 = x^2 \cdot x^2\), \(x^3 = x^2 \cdot x\), \(x^2 = x^2 \cdot 1\). GCF de variables es \(x^2\).

Step 3: Combinar GCF

GCF total es \(x^2\) (ya que GCF de coeficientes es 1, no afecta).

Step 1: Factorizar \(x^2\)

Factorizar \(x^2\) de \(x^4 - 6x^3 + 9x^2\): \(x^2(x^2 - 6x + 9)\).

Step 2: Factorizar trinomio cuadrado perfecto

\(x^2 - 6x + 9\) es \((x - 3)^2\) (ya que \((a - b)^2 = a^2 - 2ab + b^2\), con \(a = x\), \(b = 3\), \(2ab = 6x\)).

Step 3: Combinar factores

Así, \(x^4 - 6x^3 + 9x^2 = x^2(x - 3)^2\).

Answer:

\(x^2\)

Question 5: