Use synthetic division to perform each division. (-9x³ + 8x² - 7x+2) / x-2

Verified step by step guidance

1. Identify the divisor and the dividend. The dividend is the polynomial \(-9x^3 + 8x^2 - 7x + 2\), and the divisor is \(x - 2\). For synthetic division, we use the zero of the divisor, which is \$2\( (since \)x - 2 = 0\( implies \)x = 2$).
2. Write down the coefficients of the dividend polynomial in descending order of powers: \(-9\), \$8\(, \)-7\(, and \)2$.
3. Set up the synthetic division by writing the zero of the divisor (\$2$) to the left and the coefficients to the right in a row.
4. Perform synthetic division by bringing down the first coefficient, then multiply it by \$2$ and add the result to the next coefficient. Repeat this process for all coefficients.
5. Interpret the final row of numbers: the last number is the remainder, and the other numbers are the coefficients of the quotient polynomial, which will have a degree one less than the original dividend.

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. This method is efficient for finding quotients and remainders quickly.

Polynomial Coefficients

Polynomial coefficients are the numerical factors in front of the variable terms. In synthetic division, only these coefficients are used, arranged in descending order of degree. Understanding how to extract and organize these coefficients correctly is essential for applying synthetic division accurately.

Division by a Linear Binomial

Dividing by a linear binomial like x - 2 means substituting the root c = 2 into the synthetic division process. This concept is crucial because synthetic division relies on the divisor being in the form x - c, allowing the use of c in the calculation steps to find the quotient and remainder.