Question

5 multiple answer 3 points
given the function $g(x) = x^3 - 2x^2 - 5x + 6$, which statements are true? select all that apply.
$g(1)=0$, therefore $(x - 1)$ is a factor of $g(x)$.
$(x + 3)$ is a factor of $g(x)$
$(x + 2)$ is a factor of $g(x)$ because $g(x)$ divided by $(x + 2)$ has a remainder of 0
$g(1)=0$, therefore $(x + 1)$ is a factor of $g(x)$
$(x - 2)$ is a factor of $g(x)$ because $g(x)$ divided by $(x - 2)$ has a remainder of 0.
$(x - 3)$ is a factor of $g(x)$
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Explanation:

Step1: Apply Factor Theorem to (x-1)

Calculate $g(1) = 1^3 - 2(1)^2 -5(1)+6 = 1-2-5+6=0$. Since $g(1)=0$, $(x-1)$ is a factor.

Step2: Apply Factor Theorem to (x+2)

Calculate $g(-2) = (-2)^3 -2(-2)^2 -5(-2)+6 = -8-8+10+6=0$. Since $g(-2)=0$, $(x+2)$ is a factor.

Step3: Apply Factor Theorem to (x-3)

Calculate $g(3) = 3^3 -2(3)^2 -5(3)+6 = 27-18-15+6=0$. Since $g(3)=0$, $(x-3)$ is a factor.

Step4: Eliminate incorrect options

• For $(x+3)$: $g(-3)=(-3)^3-2(-3)^2-5(-3)+6=-27-18+15+6=-24

eq0$, so not a factor.

• The option linking $g(1)=0$ to $(x+1)$ is wrong, as $(x+1)$ corresponds to $g(-1)$.
• For $(x-2)$: $g(2)=2^3-2(2)^2-5(2)+6=8-8-10+6=-4

eq0$, so not a factor.

Answer:

g(1) = 0, therefore (x - 1) is a factor of g(x).
(x + 2) is a factor of g(x) because g(x) divided by (x + 2) has a remainder of 0
(x - 3) is a factor of g(x)