What is the remainder when (3x³ - 2x² + 4x - 3) is divided by (x² + 3x + 3)?

Solution:

3x³ - 2x² + 4x - 3

On division of the given polynomial, we get the remainder.

(3x³ - 2x² + 4x - 3)/(x² + 3x + 3)

This can be written as 3x³ + 9x² + 9x - 11x² - 33x - 33 + 28x + 30

Taking out the common terms

= 3x(x² + 3x + 3) - 11(x² + 3x + 3) + 28x + 30

= (x² + 3x + 3)(3x - 11) + 28x + 30

Now, (3x³ - 2x² + 4x - 3)/(x² + 3x + 3) can be re written as 

 \(\dfrac{(x^2 + 3x + 3) (3x - 11) + 28x + 30}{x^2 + 3x + 3}\)

= (3x - 11) + \(\dfrac{28x + 30}{x^2 + 3x + 3}\)

This is of the form quotient + remainder/divisor

Therefore, the remainder here is 28x + 30.

What is the remainder when (3x³ - 2x² + 4x - 3) is divided by (x² + 3x + 3)?

Summary:

The remainder when (3x³ - 2x² + 4x - 3) is divided by (x² + 3x + 3) is 28x + 30.