Divide using long division. State the quotient, and the remainder, r(x). (x⁴−81)/(x−3)

Verified step by step guidance

1. Identify the dividend and divisor. Here, the dividend is \(x^{4} - 81\) and the divisor is \(x - 3\).
2. Set up the long division by writing \(x^{4} - 81\) under the division bar and \(x - 3\) outside the bar.
3. Divide the leading term of the dividend, \(x^{4}\), by the leading term of the divisor, \(x\), to get the first term of the quotient: \(x^{3}\).
4. Multiply the entire divisor \(x - 3\) by \(x^{3}\) and subtract the result from the dividend to find the new polynomial to divide.
5. Repeat the process: divide the new leading term 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.

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, subtracting, 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 the divisor's degree.

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, and the remainder is the leftover polynomial with a degree less than the divisor, often expressed as r(x).