Use the long division method to find the result when 4, x, cubed, minus, 4, x, squared, plus, 13, x, minus, 64x 3 −4x 2 +13x−6 is divided by 2, x, minus, 12x−1. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ b(x)r(x)​ .

The given polynomial is 4x^3 - 4x^2 + 13x - 6 and we are dividing it by 2x - 1.

Step 1: Divide the first term of the numerator by the first term of the denominator i.e., 4x^3 / 2x = 2x^2. This is the first term of the quotient.

Step 2: Multiply the entire denominator by the first term of the quotient i.e., 2x^2 * (2x - 1) = 4x^3 - 2x^2.

Step 3: Subtract the result from the original polynomial i.e., (4x^3 - 4x^2 + 13x - 6) - (4x^3 - 2x^2) = -2x^2 + 13x - 6.

Step 4: Repeat the process with the new polynomial. Divide the first term of the new polynomial by the first term of the denominator i.e., -2x^2 / 2x = -x. This is the second term of the quotient.

Step 5: Multiply the entire denominator by the second term of the quotient i.e., -x * (2x - 1) = -2x^2 + x.

Step 6: Subtract the result from the new polynomial i.e., (-2x^2 + 13x - 6) - (-2x^2 + x) = 12x - 6.

Step 7: Repeat the process with the new polynomial. Divide the first term of the new polynomial by the first term of the denominator i.e., 12x / 2x = 6. This is the third term of the quotient.

Step 8: Multiply the entire denominator by the third term of the quotient i.e., 6 * (2x - 1) = 12x - 6.

Step 9: Subtract the result from the new polynomial i.e., (12x - 6) - (12x - 6) = 0. There is no remainder.

So, the quotient is 2x^2 - x + 6 and the remainder is 0. Therefore, the result of the division is 2x^2 - x + 6.