Question

1. factor. 9a² - 16
2. what does it mean when (x³ + 2x² - 12x - 24)÷(x + 2) has a remainder of zero?

Explanation:

16. Factor $9a^{2}-16$

Step1: Recognize difference - of - squares

The expression $9a^{2}-16$ is in the form $x^{2}-y^{2}$, where $x = 3a$ and $y = 4$ since $(3a)^{2}=9a^{2}$ and $4^{2}=16$.

Step2: Apply difference - of - squares formula

The formula for factoring $x^{2}-y^{2}$ is $(x + y)(x - y)$. Substituting $x = 3a$ and $y = 4$, we get $(3a+4)(3a - 4)$.

According to the factor theorem, if $f(x)$ is a polynomial and $f(c)=0$ when $f(x)$ is divided by $(x - c)$, then $(x - c)$ is a factor of $f(x)$. Here, when dividing the polynomial $x^{3}+2x^{2}-12x - 24$ by $(x + 2)$ (equivalent to $(x-(-2))$) and getting a remainder of zero, it means that $x=-2$ is a root of the polynomial $x^{3}+2x^{2}-12x - 24$, and $(x + 2)$ is a factor of the polynomial $x^{3}+2x^{2}-12x - 24$.

Answer:

$(3a + 4)(3a-4)$

18. What does it mean when $(x^{3}+2x^{2}-12x - 24)\div(x + 2)$ has a remainder of zero?