10 multiple choice 1 point supposed ( g(x) = 6x^3 - ax^2 + 2x + 4 ), where ( a ) is an unknown real number. if the remainder of the division of ( g(x) ) by ( (x - 2) ) is 44, then what is the remainder when dividing by ( (x + 3) )?
options: (-191), (3), (-137), (145)

Question

Accepted Answer

# Explanation:
## Step1: Use Remainder Theorem for $x - 2$
The Remainder Theorem states that the remainder of dividing a polynomial $g(x)$ by $x - c$ is $g(c)$. For $g(x)$ divided by $x - 2$, the remainder is $g(2)$. So we calculate $g(2)$:
$$
\begin{align*}
g(2)&=6(2)^3 - a(2)^2 + 2(2) + 4\
&=6\times8 - a\times4 + 4 + 4\
&=48 - 4a + 8\
&=56 - 4a
\end{align*}
$$
We know this remainder is 44, so:
$$
56 - 4a = 44
$$
## Step2: Solve for $a$
Subtract 56 from both sides:
$$
-4a = 44 - 56\
-4a = -12
$$
Divide both sides by -4:
$$
a = \frac{-12}{-4} = 3
$$
## Step3: Find $g(x)$ with $a = 3$
Now that $a = 3$, the polynomial $g(x)$ is:
$$
g(x)=6x^3 - 3x^2 + 2x + 4
$$
## Step4: Use Remainder Theorem for $x + 3$
To find the remainder when dividing by $x + 3$, we use the Remainder Theorem with $c = -3$ (since $x + 3 = x - (-3)$). So we calculate $g(-3)$:
$$
\begin{align*}
g(-3)&=6(-3)^3 - 3(-3)^2 + 2(-3) + 4\
&=6\times(-27) - 3\times9 - 6 + 4\
&=-162 - 27 - 6 + 4\
&=-162 - 27 = -189; -189 - 6 = -195; -195 + 4 = -191
\end{align*}
$$
# Answer:
$-191$ (corresponding to the option: $-191$)

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QUESTION IMAGE

Question

Explanation:

Step1: Use Remainder Theorem for \(x - 2\)

Step2: Solve for \(a\)

Step3: Find \(g(x)\) with \(a = 3\)

Step4: Use Remainder Theorem for \(x + 3\)

Answer: