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

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To find the quotient of the polynomials represented by the coefficients 6, 15, 10, 10, and 4 at varying powers of x, we need to divide the first polynomial by the second polynomial. Let’s assume the two polynomials are:

  • Numerator: 6x4 + 15x3 + 10x2 + 10x + 4
  • Denominator: 3x2 + 2

We will perform polynomial long division to find this quotient.

Step 1: Divide the leading terms

Take the leading term of the numerator, which is 6x4, and divide it by the leading term of the denominator, which is 3x2. This results in:

6x^4 ÷ 3x^2 = 2x^2

Step 2: Multiply and subtract

Now multiply the entire denominator by 2x2:

(3x^2 + 2) * 2x^2 = 6x^4 + 4x^2

Next, subtract this from the original numerator:

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

Step 3: Repeat the process

Now, take the new polynomial, 15x3 + 6x2 + 10x + 4, and repeat the division with the leading terms:

15x^3 ÷ 3x^2 = 5x

Multiply the entire denominator:

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

Subtract again:

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

Step 4: Final division

Now we divide 6x2 by 3x2: 

6x^2 ÷ 3x^2 = 2

Multiply the denominator:

(3x^2 + 2) * 2 = 6x^2 + 4

When we subtract this from the polynomial:

(6x^2 + 4) - (6x^2 + 4) = 0

Final Result

Since everything cancels out, the final quotient of the given polynomials is:

Q(x) = 2x^2 + 5x + 2

In conclusion, the quotient of the polynomials 6x4, 15x3, 10x2, 10x, and 4 divided by 3x2 + 2 is:

2x^2 + 5x + 2

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