Answer :
To find the remainder when dividing the polynomial [tex]\( f(x) = 3x^3 + 2x^2 - 3x + 8 \)[/tex] by [tex]\( x + 4 \)[/tex], we can use the Remainder Theorem. According to the Remainder Theorem, when a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - c \)[/tex], the remainder is equal to [tex]\( f(c) \)[/tex].
In this problem, we are dividing by [tex]\( x + 4 \)[/tex], which can be rewritten as [tex]\( x - (-4) \)[/tex]. Hence, the value of [tex]\( c \)[/tex] is [tex]\(-4\)[/tex].
To find the remainder, we need to evaluate [tex]\( f(-4) \)[/tex]:
1. Substitute [tex]\( x = -4 \)[/tex] into the polynomial [tex]\( f(x) \)[/tex].
[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]
- [tex]\( 8 = 8 \)[/tex]
3. Add these results together:
[tex]\[
f(-4) = -192 + 32 + 12 + 8
\][/tex]
4. Perform the addition:
[tex]\[
-192 + 32 = -160
\][/tex]
[tex]\[
-160 + 12 = -148
\][/tex]
[tex]\[
-148 + 8 = -140
\][/tex]
Therefore, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 4 \)[/tex] is [tex]\(-140\)[/tex]. The correct answer is (B) [tex]\( f(-4) = -140 \)[/tex].
In this problem, we are dividing by [tex]\( x + 4 \)[/tex], which can be rewritten as [tex]\( x - (-4) \)[/tex]. Hence, the value of [tex]\( c \)[/tex] is [tex]\(-4\)[/tex].
To find the remainder, we need to evaluate [tex]\( f(-4) \)[/tex]:
1. Substitute [tex]\( x = -4 \)[/tex] into the polynomial [tex]\( f(x) \)[/tex].
[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]
- [tex]\( 8 = 8 \)[/tex]
3. Add these results together:
[tex]\[
f(-4) = -192 + 32 + 12 + 8
\][/tex]
4. Perform the addition:
[tex]\[
-192 + 32 = -160
\][/tex]
[tex]\[
-160 + 12 = -148
\][/tex]
[tex]\[
-148 + 8 = -140
\][/tex]
Therefore, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 4 \)[/tex] is [tex]\(-140\)[/tex]. The correct answer is (B) [tex]\( f(-4) = -140 \)[/tex].