Dividing Polynomials

To find the remainder when one polynomial is divided by another, we can use polynomial long division. In this case, we want to divide the polynomial 6X^4 + 8X^3 + 17X^2 + 21X + 7 by the polynomial 3X^2 + 4X + 1.

Polynomial Long Division

The process of polynomial long division is similar to regular long division of numbers. We start by dividing the highest degree term of the dividend (6X^4) by the highest degree term of the divisor (3X^2). This gives us 2X^2 as the first term of the quotient.

Next, we multiply the entire divisor (3X^2 + 4X + 1) by the first term of the quotient (2X^2) and subtract the result from the dividend (6X^4 + 8X^3 + 17X^2 + 21X + 7). This gives us a new polynomial to work with.

After the subtraction, we have a new polynomial: 2X^2 + 8X + 6.

Continuing the Division

We repeat the process by dividing the highest degree term of the new polynomial (2X^2) by the highest degree term of the divisor (3X^2). This gives us another term for the quotient, which is 2/3 of X.

Next, we multiply the entire divisor (3X^2 + 4X + 1) by the new term of the quotient (2/3X) and subtract the result from the new polynomial (2X^2 + 8X + 6). This gives us another new polynomial to work with.

After the subtraction, we have a new polynomial: (2/3X + 8/3)X + 10/3.

Final Steps

We repeat the process one more time by dividing the highest degree term of the new polynomial ((2/3)X + 8/3) by the highest degree term of the divisor (3X^2). This gives us the final term of the quotient, which is (2/9)X.

Next, we multiply the entire divisor (3X^2 + 4X + 1) by the final term of the quotient (2/9)X and subtract the result from the new polynomial ((2/3)X + 8/3)X + 10/3. This gives us the remainder of the division.

After the subtraction, we have a remainder of (-14/9)X + (25/3).

Conclusion

The remainder when the polynomial 6X^4 + 8X^3 + 17X^2 + 21X + 7 is divided by the polynomial 3X^2 + 4X + 1 is (-14/9)X + (25/3).

Therefore, the values of a and b are -14/9 and 25/3, respectively.