In Exercises 17–32, divide using synthetic division. (x⁴−256)/(x−4)

Verified step by step guidance

1. Identify the divisor and the dividend. Here, the divisor is $x-4$ and the dividend is $x{x}^4-256$.
2. Write down the coefficients of the dividend polynomial in descending order of powers of $x$. Since the polynomial is $x^4\; -\; 256$, the coefficients are $1$ (for $x^4$), $0$ (for $x^3$), $0$ (for $x^2$), $0$ (for $x$), and $-256$ (constant term).
3. Set up synthetic division by writing the zero of the divisor, which is $4$, to the left. Then write the coefficients of the dividend to the right in a row: $1,\; 0,\; 0,\; 0,\; -256$.
4. Perform synthetic division: bring down the first coefficient, multiply it by 4, write the result under the next coefficient, add down the column, and repeat this process for all coefficients.
5. Interpret the final row of numbers as the coefficients of the quotient polynomial, starting from one degree less than the original dividend (so degree 3), and the last number as the remainder.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - c). It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone.

Polynomial Coefficients and Degree

Understanding the coefficients and degree of a polynomial is essential for synthetic division. The degree indicates the highest power of x, and coefficients are the numerical factors of each term, which are used in the synthetic division setup.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is f(c). This concept helps verify the result of synthetic division and understand the relationship between division and polynomial evaluation.