College

What is the remainder when [tex]3x^3 - 2x^2 + 4x - 3[/tex] is divided by [tex]3x - 11[/tex]?

A. 30
B. [tex]28x - 36[/tex]
C. [tex]28x + 30[/tex]

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]