Use synthetic division to perform each division. (x⁴ - 3x³ - 4x² + 12x) / x-2

Verified step by step guidance

1. Identify the divisor and the dividend. The divisor is \(x - 2\), so the root to use in synthetic division is \(2\) (since \(x - 2 = 0\) implies \(x = 2\)). The dividend is \(x^4 - 3x^3 - 4x^2 + 12x\).
2. Write down the coefficients of the dividend in descending order of powers of \(x\). For \(x^4 - 3x^3 - 4x^2 + 12x + 0\), the coefficients are \([1, -3, -4, 12, 0]\). Note the \(0\) for the constant term since it is missing.
3. Set up the synthetic division by placing the root \(2\) to the left and the coefficients to the right. Begin the process by bringing down the first coefficient \(1\) as is.
4. Multiply the root \(2\) by the number just written below the line, then write the result under the next coefficient. Add the column and write the sum below the line. Repeat this multiply-and-add process for all coefficients.
5. After completing the synthetic division, interpret the bottom row 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 and performing arithmetic operations in a tabular form. This method is efficient for finding quotients and remainders quickly.

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 represented with zero coefficients. For example, in x^4 - 3x^3 - 4x^2 + 12x, the constant term is zero and should be included as such.

Division by a Linear Binomial (x - c)

Dividing by a linear binomial like x - 2 means substituting c = 2 in synthetic division. This value is used to perform the synthetic division steps, which helps determine the quotient polynomial and remainder. Recognizing the divisor's form is crucial to apply synthetic division correctly.