College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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]