Divide using long division. State the quotient, and the remainder, r(x). (4x⁴−4x²+6x)/(x−4)

Verified step by step guidance

1. Write the division in long division format, placing the dividend \$4x^{4} - 4x^{2} + 6x\( under the division bar and the divisor \)x - 4$ outside.
2. Identify the first term of the quotient by dividing the leading term of the dividend \$4x^{4}\( by the leading term of the divisor \)x\(, which gives \)4x^{3}$.
3. Multiply the entire divisor \(x - 4\) by this term \$4x^{3}\(, resulting in \)4x^{4} - 16x^{3}$, and subtract this from the dividend to find the new remainder.
4. Bring down the next terms and repeat the process: divide the leading term of the new remainder by \(x\), multiply the divisor by this term, subtract, and continue until the degree of the remainder is less than the degree of the divisor.
5. Express the final answer as the quotient polynomial plus the remainder over the divisor, in the form \(\text{quotient} + \frac{r(x)}{x - 4}\).

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by this result, subtracting from the dividend, and repeating until the degree of the remainder is less than the divisor.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. Understanding the degree helps determine when to stop the division process, as the remainder must have a degree less than that of the divisor for the division to be complete.

Quotient and Remainder in Polynomial Division

When dividing polynomials, the result consists of a quotient and a remainder. The quotient is the polynomial obtained from the division process, and the remainder is the leftover polynomial with a degree less than the divisor. Expressing the division as dividend = divisor × quotient + remainder is essential.