Answer :
To find the remainder when [tex]\( f(x) = 2x^3 + 3x^2 - 11x - 6 \)[/tex] is divided by [tex]\( x-3 \)[/tex], we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - c \)[/tex], the remainder of that division is [tex]\( f(c) \)[/tex].
Here, we are given [tex]\( f(x) = 2x^3 + 3x^2 - 11x - 6 \)[/tex] and want to find the remainder when divided by [tex]\( x - 3 \)[/tex]. This means we need to evaluate [tex]\( f(3) \)[/tex].
Let's substitute [tex]\( x = 3 \)[/tex] into the polynomial:
1. Compute [tex]\( 2(3)^3 \)[/tex]:
[tex]\[
2 \times 27 = 54
\][/tex]
2. Compute [tex]\( 3(3)^2 \)[/tex]:
[tex]\[
3 \times 9 = 27
\][/tex]
3. Compute [tex]\( -11(3) \)[/tex]:
[tex]\[
-11 \times 3 = -33
\][/tex]
4. Finally, add the constant term [tex]\(-6\)[/tex].
Now, sum all these results together:
[tex]\[
f(3) = 54 + 27 - 33 - 6
\][/tex]
Calculating this gives:
[tex]\[
54 + 27 = 81
\][/tex]
[tex]\[
81 - 33 = 48
\][/tex]
[tex]\[
48 - 6 = 42
\][/tex]
So, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x-3 \)[/tex] is [tex]\( 42 \)[/tex].
The correct answer is:
(D) [tex]\( f(3) = 42 \)[/tex]
Here, we are given [tex]\( f(x) = 2x^3 + 3x^2 - 11x - 6 \)[/tex] and want to find the remainder when divided by [tex]\( x - 3 \)[/tex]. This means we need to evaluate [tex]\( f(3) \)[/tex].
Let's substitute [tex]\( x = 3 \)[/tex] into the polynomial:
1. Compute [tex]\( 2(3)^3 \)[/tex]:
[tex]\[
2 \times 27 = 54
\][/tex]
2. Compute [tex]\( 3(3)^2 \)[/tex]:
[tex]\[
3 \times 9 = 27
\][/tex]
3. Compute [tex]\( -11(3) \)[/tex]:
[tex]\[
-11 \times 3 = -33
\][/tex]
4. Finally, add the constant term [tex]\(-6\)[/tex].
Now, sum all these results together:
[tex]\[
f(3) = 54 + 27 - 33 - 6
\][/tex]
Calculating this gives:
[tex]\[
54 + 27 = 81
\][/tex]
[tex]\[
81 - 33 = 48
\][/tex]
[tex]\[
48 - 6 = 42
\][/tex]
So, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x-3 \)[/tex] is [tex]\( 42 \)[/tex].
The correct answer is:
(D) [tex]\( f(3) = 42 \)[/tex]
