Divide using synthetic division. (6x⁵−2x³+4x²−3x+1)÷(x−2)

Verified step by step guidance

1. Identify the dividend polynomial and the divisor. Here, the dividend is \$6x^{5} - 2x^{3} + 4x^{2} - 3x + 1\( and the divisor is \)x - 2$.
2. Set up synthetic division by writing the coefficients of the dividend polynomial in descending order of powers of \(x\). Since the polynomial is missing the \(x^{4}\) term, include a 0 for its coefficient. The coefficients are: \$6, 0, -2, 4, -3, 1$.
3. Write the zero of the divisor \(x - 2\) which is \$2\( (since \)x - 2 = 0\( implies \)x = 2$) to the left of the synthetic division setup.
4. Perform synthetic division by bringing down the first coefficient, then multiply it by 2 and add to the next coefficient, 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 polynomial, 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 and performing arithmetic operations in a tabular form.

Polynomial Coefficients and Terms

Understanding the coefficients and terms of a polynomial is essential for synthetic division. Each term's coefficient is used in the division process, and missing degrees must be accounted for by inserting zero coefficients to maintain proper alignment.

Remainder and Quotient Interpretation

After performing synthetic division, the resulting numbers represent the coefficients of the quotient polynomial, and the last number is the remainder. Interpreting these correctly helps in expressing the division result as quotient plus remainder over the divisor.