What is the remainder when $(3 x^3-2 x^2+4 x-3)$ is divided by $(x^2+3 x+3)$?

Perform polynomial long division of 3 x 3 − 2 x 2 + 4 x − 3 by x 2 + 3 x + 3 .
The quotient is 3 x − 11 and the remainder is 28 x + 30 .
Verify the result: ( x 2 + 3 x + 3 ) ( 3 x − 11 ) + ( 28 x + 30 ) = 3 x 3 − 2 x 2 + 4 x − 3 .
The remainder is 28 x + 30 ​ .

Explanation

Problem Analysis We are given the polynomial 3 x 3 − 2 x 2 + 4 x − 3 and asked to find the remainder when it is divided by x 2 + 3 x + 3 .

Solution Strategy We will perform polynomial long division to find the quotient and remainder. The remainder will be of the form a x + b since we are dividing by a quadratic.

Polynomial Long Division Performing polynomial long division of 3 x 3 − 2 x 2 + 4 x − 3 by x 2 + 3 x + 3 , we obtain the quotient 3 x − 11 and the remainder 28 x + 30 . We can verify this by multiplying the quotient by the divisor and adding the remainder:


( x 2 + 3 x + 3 ) ( 3 x − 11 ) + ( 28 x + 30 ) = 3 x 3 − 11 x 2 + 9 x 2 − 33 x + 9 x − 33 + 28 x + 30 = 3 x 3 − 2 x 2 + 4 x − 3 .

Finding the Remainder Therefore, the remainder is 28 x + 30 .

Examples
Polynomial division is a fundamental concept in algebra and is used in various applications, such as simplifying rational expressions, solving polynomial equations, and analyzing the behavior of polynomial functions. For instance, in engineering, polynomial division can be used to model the trajectory of a projectile or to analyze the stability of a control system. By understanding polynomial division, students can gain a deeper appreciation for the power and versatility of algebraic techniques in solving real-world problems.

Answered by GinnyAnswer | 2025-07-07