Question

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)

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: \(-191\))