In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (x⁴−81)/(x−3)

Verified step by step guidance

1. Set up the long division by writing the dividend $x^4-81$ under the division bar and the divisor $x\; -\; 3$ outside the division bar.
2. 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: $x^3$.
3. Multiply the entire divisor $(x\; -\; 3)$ by this term $x^3$ and subtract the result from the dividend to find the new remainder.
4. Repeat the process: divide the leading term of the new remainder by $x$, multiply the divisor by this term, subtract again, 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 $Q(x)\; +\; \backslash frac\{r(x)\}\{x\; -\; 3\}$.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree, 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 remainder has a lower degree than the divisor.

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. The division can be expressed as dividend = divisor × quotient + remainder.

Difference of Squares

The difference of squares is a special factoring formula: a² - b² = (a - b)(a + b). Recognizing this pattern helps simplify polynomials like x⁴ - 81, since 81 is 9² and x⁴ is (x²)², allowing factorization into (x² - 9)(x² + 9) before or after division.