Perform $(3 x^3-2 x^2+3 x-4) \div(x-3)$ to find the value of the remainder.
A) 88
B) 78
C) 98
D) 68

Apply the Remainder Theorem: The remainder when dividing a polynomial P ( x ) by ( x − c ) is P ( c ) .
Substitute x = 3 into the polynomial: P ( 3 ) = 3 ( 3 ) 3 − 2 ( 3 ) 2 + 3 ( 3 ) − 4 .
Calculate the value: P ( 3 ) = 81 − 18 + 9 − 4 = 68 .
The remainder is 68 ​ .

Explanation

Problem Analysis We are given the polynomial P ( x ) = 3 x 3 − 2 x 2 + 3 x − 4 and we want to divide it by ( x − 3 ) to find the remainder.

Remainder Theorem According to the Remainder Theorem, if we divide a polynomial P ( x ) by ( x − c ) , the remainder is P ( c ) . In our case, we are dividing by ( x − 3 ) , so c = 3 .

Evaluating the Polynomial We need to evaluate the polynomial P ( x ) at x = 3 . That is, we need to find P ( 3 ) . P ( 3 ) = 3 ( 3 ) 3 − 2 ( 3 ) 2 + 3 ( 3 ) − 4

Calculating the Remainder Now, let's calculate P ( 3 ) : P ( 3 ) = 3 ( 27 ) − 2 ( 9 ) + 3 ( 3 ) − 4 P ( 3 ) = 81 − 18 + 9 − 4 P ( 3 ) = 63 + 9 − 4 P ( 3 ) = 72 − 4 P ( 3 ) = 68

Final Answer Therefore, the remainder when 3 x 3 − 2 x 2 + 3 x − 4 is divided by ( x − 3 ) is 68.


Examples
Understanding polynomial remainders is crucial in various fields, such as computer science for error detection codes and in engineering for system analysis. For example, if you're designing a communication system and need to ensure data integrity, the remainder theorem helps in creating efficient error-checking algorithms. By representing data as polynomials, remainders can indicate whether data packets have been corrupted during transmission, allowing for immediate correction or retransmission, ensuring reliable communication.

Answered by GinnyAnswer | 2025-07-08

The remainder when dividing the polynomial 3 x 3 − 2 x 2 + 3 x − 4 by ( x − 3 ) is 68 . Thus, the correct answer is option D) 68.
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Answered by Anonymous | 2025-08-22