Answer :
We start with the Division Algorithm for polynomials. It tells us that when a polynomial [tex]\( f(x) \)[/tex] is divided by a divisor [tex]\( d(x) \)[/tex], there exist a quotient [tex]\( q(x) \)[/tex] and a remainder [tex]\( r(x) \)[/tex] such that
[tex]$$
f(x) = q(x) \, d(x) + r(x),
$$[/tex]
and the degree of [tex]\( r(x) \)[/tex] is less than the degree of [tex]\( d(x) \)[/tex].
In this problem, the given polynomial is
[tex]$$
f(x) = 3x^3 - 2x^2 + 4x - 3.
$$[/tex]
Notice that the divisor provided in the answer choices is a linear polynomial (degree 1). This means that the remainder [tex]\( r(x) \)[/tex] must be a constant (a polynomial of degree 0).
Now, among the choices:
- [tex]\( 30 \)[/tex] is a constant.
- [tex]\( 3x - 11 \)[/tex], [tex]\( 28x - 36 \)[/tex], and [tex]\( 28x + 30 \)[/tex] are all linear polynomials.
Since the remainder must be a constant, the only viable option is [tex]\( 30 \)[/tex].
Thus, the remainder when [tex]\( 3x^3 - 2x^2 + 4x - 3 \)[/tex] is divided by any linear divisor is
[tex]$$
\boxed{30}.
$$[/tex]
[tex]$$
f(x) = q(x) \, d(x) + r(x),
$$[/tex]
and the degree of [tex]\( r(x) \)[/tex] is less than the degree of [tex]\( d(x) \)[/tex].
In this problem, the given polynomial is
[tex]$$
f(x) = 3x^3 - 2x^2 + 4x - 3.
$$[/tex]
Notice that the divisor provided in the answer choices is a linear polynomial (degree 1). This means that the remainder [tex]\( r(x) \)[/tex] must be a constant (a polynomial of degree 0).
Now, among the choices:
- [tex]\( 30 \)[/tex] is a constant.
- [tex]\( 3x - 11 \)[/tex], [tex]\( 28x - 36 \)[/tex], and [tex]\( 28x + 30 \)[/tex] are all linear polynomials.
Since the remainder must be a constant, the only viable option is [tex]\( 30 \)[/tex].
Thus, the remainder when [tex]\( 3x^3 - 2x^2 + 4x - 3 \)[/tex] is divided by any linear divisor is
[tex]$$
\boxed{30}.
$$[/tex]
