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Answer :
Sure! Let's use the Remainder Theorem to find the remainder when the polynomial [tex]\( f(x) = 3x^3 + 2x^2 - 3x + 8 \)[/tex] is divided by [tex]\( x + 4 \)[/tex].
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]. Here, our divisor is [tex]\( x + 4 \)[/tex], which can be rewritten as [tex]\( x - (-4) \)[/tex]. Thus, [tex]\( c = -4 \)[/tex].
To find the remainder, we need to evaluate the polynomial at [tex]\( x = -4 \)[/tex].
Let's substitute [tex]\( x = -4 \)[/tex] into the polynomial:
1. First, calculate [tex]\( (-4)^3 \)[/tex]:
[tex]\((-4) \times (-4) \times (-4) = -64\)[/tex]
So, [tex]\( 3 \times (-64) = -192 \)[/tex].
2. Next, calculate [tex]\( (-4)^2 \)[/tex]:
[tex]\((-4) \times (-4) = 16\)[/tex]
So, [tex]\( 2 \times 16 = 32 \)[/tex].
3. Then, calculate [tex]\(-3 \times (-4)\)[/tex]:
This equals [tex]\( 12 \)[/tex].
4. Finally, add the constant term [tex]\( 8 \)[/tex].
Now, add all these results together:
[tex]\[
-192 + 32 + 12 + 8 = -140
\][/tex]
Thus, the remainder when the polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 4 \)[/tex] is [tex]\(-140\)[/tex].
Therefore, the correct answer is (B) [tex]\( f(-4) = -140 \)[/tex].
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]. Here, our divisor is [tex]\( x + 4 \)[/tex], which can be rewritten as [tex]\( x - (-4) \)[/tex]. Thus, [tex]\( c = -4 \)[/tex].
To find the remainder, we need to evaluate the polynomial at [tex]\( x = -4 \)[/tex].
Let's substitute [tex]\( x = -4 \)[/tex] into the polynomial:
1. First, calculate [tex]\( (-4)^3 \)[/tex]:
[tex]\((-4) \times (-4) \times (-4) = -64\)[/tex]
So, [tex]\( 3 \times (-64) = -192 \)[/tex].
2. Next, calculate [tex]\( (-4)^2 \)[/tex]:
[tex]\((-4) \times (-4) = 16\)[/tex]
So, [tex]\( 2 \times 16 = 32 \)[/tex].
3. Then, calculate [tex]\(-3 \times (-4)\)[/tex]:
This equals [tex]\( 12 \)[/tex].
4. Finally, add the constant term [tex]\( 8 \)[/tex].
Now, add all these results together:
[tex]\[
-192 + 32 + 12 + 8 = -140
\][/tex]
Thus, the remainder when the polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 4 \)[/tex] is [tex]\(-140\)[/tex].
Therefore, the correct answer is (B) [tex]\( f(-4) = -140 \)[/tex].
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