Answer :
To find the remainder when [tex]\( f(x) = 3x^3 + 2x^2 - 3x + 8 \)[/tex] is divided by [tex]\( x + 4 \)[/tex], we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial [tex]\( f(x) \)[/tex] by a linear divisor [tex]\( x - c \)[/tex] is [tex]\( f(c) \)[/tex].
In this case, our divisor is [tex]\( x + 4 \)[/tex], which can be written as [tex]\( x - (-4) \)[/tex]. According to the Remainder Theorem, the remainder is [tex]\( f(-4) \)[/tex].
Let's evaluate [tex]\( f(-4) \)[/tex]:
1. Substitute [tex]\(-4\)[/tex] into the polynomial:
[tex]\[
f(-4) = 3(-4)^3 + 2(-4)^2 - 3(-4) + 8
\][/tex]
2. Calculate each term:
- [tex]\(3(-4)^3 = 3 \times (-64) = -192\)[/tex]
- [tex]\(2(-4)^2 = 2 \times 16 = 32\)[/tex]
- [tex]\(-3(-4) = 12\)[/tex]
- The constant term is [tex]\(8\)[/tex].
3. Add these values together:
[tex]\[
f(-4) = -192 + 32 + 12 + 8
\][/tex]
4. Simplify the expression:
[tex]\[
-192 + 32 = -160
\][/tex]
[tex]\[
-160 + 12 = -148
\][/tex]
[tex]\[
-148 + 8 = -140
\][/tex]
So, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 4 \)[/tex] is [tex]\(-140\)[/tex].
Therefore, the correct answer is [tex]\( \boxed{\text{(B) } f(-4) = -140} \)[/tex].
In this case, our divisor is [tex]\( x + 4 \)[/tex], which can be written as [tex]\( x - (-4) \)[/tex]. According to the Remainder Theorem, the remainder is [tex]\( f(-4) \)[/tex].
Let's evaluate [tex]\( f(-4) \)[/tex]:
1. Substitute [tex]\(-4\)[/tex] into the polynomial:
[tex]\[
f(-4) = 3(-4)^3 + 2(-4)^2 - 3(-4) + 8
\][/tex]
2. Calculate each term:
- [tex]\(3(-4)^3 = 3 \times (-64) = -192\)[/tex]
- [tex]\(2(-4)^2 = 2 \times 16 = 32\)[/tex]
- [tex]\(-3(-4) = 12\)[/tex]
- The constant term is [tex]\(8\)[/tex].
3. Add these values together:
[tex]\[
f(-4) = -192 + 32 + 12 + 8
\][/tex]
4. Simplify the expression:
[tex]\[
-192 + 32 = -160
\][/tex]
[tex]\[
-160 + 12 = -148
\][/tex]
[tex]\[
-148 + 8 = -140
\][/tex]
So, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 4 \)[/tex] is [tex]\(-140\)[/tex].
Therefore, the correct answer is [tex]\( \boxed{\text{(B) } f(-4) = -140} \)[/tex].