In Exercises 17–32, divide using synthetic division. (6x⁵−2x³+4x²−3x+1)÷(x−2)

Verified step by step guidance

1. Identify the dividend and divisor. The dividend is the polynomial $\mathrm{6x^5}-2\mathrm{x^3}+4\mathrm{x^2}-3x+1$ and the divisor is $x-2$.
2. Set up synthetic division by writing the coefficients of the dividend in descending order of powers of $x$. Since the polynomial is $6x^5\; -\; 2x^3\; +\; 4x^2\; -\; 3x\; +\; 1$, note that the coefficient for $x^4$ is missing, so include a 0 for that term. The coefficients are: $6,\; 0,\; -2,\; 4,\; -3,\; 1$.
3. Write the zero of the divisor $x\; -\; 2$, which is $2$, to the left of the synthetic division setup.
4. Perform synthetic division by bringing down the first coefficient, multiplying it by 2, adding 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 polynomials, making calculations faster and less error-prone.

Polynomial Coefficients and Terms

Understanding polynomial coefficients and terms is essential for synthetic division. Each term's coefficient is used in the synthetic division process, and missing terms must be accounted for with zero coefficients to maintain the correct degree sequence.

Remainder Theorem

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