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Answer :
To find the remainder when the polynomial [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 if a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - a \)[/tex], the remainder of this division is [tex]\( f(a) \)[/tex].
In this problem, we are dividing by [tex]\( x + 4 \)[/tex], which can be rewritten as [tex]\( x - (-4) \)[/tex]. Therefore, according to the Remainder Theorem, the remainder is equal to [tex]\( f(-4) \)[/tex].
Let's evaluate [tex]\( f(-4) \)[/tex]:
1. Start with the polynomial [tex]\( f(x) = 3x^3 + 2x^2 - 3x + 8 \)[/tex].
2. Substitute [tex]\( x = -4 \)[/tex] into the polynomial:
[tex]\[
f(-4) = 3(-4)^3 + 2(-4)^2 - 3(-4) + 8
\][/tex]
3. 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]
4. Add these values together:
[tex]\[
f(-4) = -192 + 32 + 12 + 8
\][/tex]
5. Simplifying the expression, we find:
[tex]\[
f(-4) = -140
\][/tex]
Therefore, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x+4 \)[/tex] is [tex]\(-140\)[/tex]. The correct choice 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]. Therefore, according to the Remainder Theorem, the remainder is equal to [tex]\( f(-4) \)[/tex].
Let's evaluate [tex]\( f(-4) \)[/tex]:
1. Start with the polynomial [tex]\( f(x) = 3x^3 + 2x^2 - 3x + 8 \)[/tex].
2. Substitute [tex]\( x = -4 \)[/tex] into the polynomial:
[tex]\[
f(-4) = 3(-4)^3 + 2(-4)^2 - 3(-4) + 8
\][/tex]
3. 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]
4. Add these values together:
[tex]\[
f(-4) = -192 + 32 + 12 + 8
\][/tex]
5. Simplifying the expression, we find:
[tex]\[
f(-4) = -140
\][/tex]
Therefore, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x+4 \)[/tex] is [tex]\(-140\)[/tex]. The correct choice is (B) [tex]\( f(-4) = -140 \)[/tex].
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