In Exercises 17–32, divide using synthetic division. (x⁵+x³−2)/(x−1)

Verified step by step guidance

1. Identify the dividend and divisor. The dividend is the polynomial $x^5+x^3-\; 2$ and the divisor is $x\; -\; 1$.
2. Set up synthetic division by writing the coefficients of the dividend in descending order of powers of $x$. Since the polynomial is missing the $x^4$ and $x^2$ terms, include zeros for those coefficients. The coefficients are: 1 (for $x^5$), 0 (for $x^4$), 1 (for $x^3$), 0 (for $x^2$), 0 (for $x$), and -2 (constant term).
3. Write the zero of the divisor $x\; -\; 1$ which is $1$ to the left of the synthetic division setup.
4. Perform synthetic division by bringing down the first coefficient, multiplying it by the zero of the divisor, adding the result to the next coefficient, and repeating this process across 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, 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 degree of a polynomial and its coefficients is essential for synthetic division. The degree indicates the highest power of x, and all terms must be accounted for, including those with zero coefficients, to correctly set up the synthetic division.

Remainder and Quotient Interpretation

After performing synthetic division, the result includes a quotient polynomial and possibly a remainder. Interpreting these correctly helps in understanding the division outcome, where the remainder is the constant left over and the quotient is the polynomial result of the division.