What is the quotient of 6x^4, 15x^3, 2x^2, 10x, and 4, 3x^2, 2?

To find the quotient of the given polynomials, we first need to clarify what we mean by ‘quotient.’ Here, we are looking at the expression:

• Numerator: 6x^4 + 15x^3 + 2x^2 + 10x + 4
• Denominator: 3x^2 + 2

Next, we will divide the numerator by the denominator using polynomial long division. Here’s how:

Step-by-Step Division:

1. Divide the first term of the numerator (6x^4) by the first term of the denominator (3x^2), which gives us 2x^2.
2. Multiply the entire denominator (3x^2 + 2) by this result (2x^2) giving us 6x^4 + 4x^2.
3. Subtract this from the original numerator:

   (6x^4 + 15x^3 + 2x^2 + 10x + 4)
-  (6x^4 + 4x^2)
  --------------------------------
         (15x^3 - 2x^2 + 10x + 4)

4. Now, repeat the process: divide 15x^3 by 3x^2 to get 5x.
5. Multiply the denominator by 5x:

  (3x^2 + 2) * 5x = 15x^3 + 10x

6. Subtract again:

   (15x^3 - 2x^2 + 10x + 4)
-  (15x^3 + 10x)
  ----------------------------------
            (-2x^2 + 4)

7. Now, divide -2x^2 by 3x^2 to get -2/3.
8. Multiply the entire denominator by -2/3:

-2/3 * (3x^2 + 2) = -2x^2 - 4/3

9. Subtract once more:

   (-2x^2 + 4)
- (-2x^2 - 4/3)
  ---------------------
       (4 + 4/3) = (12/3 + 4/3) = (16/3)

This means the final quotient is:

2x^2 + 5x - 2/3 with a remainder of 16/3

Therefore, the result of dividing 6x^4 + 15x^3 + 2x^2 + 10x + 4 by 3x^2 + 2 yields:

2x^2 + 5x – 2/3 + (16/3)/(3x^2 + 2)