Final answer:
The equation (4x ^ 4 - 2x ^ 3 - 13x ^ 2 + 8x - 9)/(x ^ 2 + 3) is solved using long division, which involves dividing the polynomial by x^2 + 3 to find the quotient. Step-by-step, we subtract the divisor multiplied by the determined quotient from the dividend until the remainder is of a lower degree than the divisor.
Step-by-step explanation:
The student has asked to solve the expression (4x ^ 4 - 2x ^ 3 - 13x ^ 2 + 8x - 9)/(x ^ 2 + 3) using long division to find the quotient. Unfortunately, the provided references do not directly correspond to the question asked; therefore, the long division method must be explained without them. Long division in algebra is similar to long division with numbers. It involves determining how many times the divisor, in this case, x2 + 3, goes into the dividend, which is the polynomial 4x4 - 2x3 - 13x2 + 8x - 9. Simply put, we divide the first term of the dividend by the first term of the divisor to find each term of the quotient and subtract the result from the dividend repeatedly until we reach a remainder that cannot be divided further by the divisor.
To perform the division, we write down the dividend and divisor similarly to numerical long division and find how many times the highest degree term of the divisor, x2, fits into the highest degree term of the dividend, 4x4. We then multiply the whole divisor by that figure and subtract the result from the dividend. We repeat this process, bringing down terms as needed until the degree of the remainder is less than the degree of the divisor. The quotient thus obtained will be the result of the division.