In Exercises 17–32, divide using synthetic division. (6x^5−2x^3+4x^2−3x+1)÷(x−2)

Verified step by step guidance

1. Identify the divisor and set it equal to zero to find the value for synthetic division. Here, the divisor is \(x - 2\), so set \(x - 2 = 0\) to find \(x = 2\).
2. Write down the coefficients of the dividend \(6x^5 - 2x^3 + 4x^2 - 3x + 1\). Note that there is no \(x^4\) term, so its coefficient is 0. The coefficients are: \([6, 0, -2, 4, -3, 1]\).
3. Set up the synthetic division by writing the value \(2\) from step 1 on the left and the coefficients from step 2 in a row to the right.
4. Bring down the first coefficient \(6\) as it is. Multiply it by \(2\) and write the result under the next coefficient. Add this result to the next coefficient and write the sum below.
5. Repeat the multiplication and addition process for each coefficient until you reach the end. The final row of numbers represents the coefficients of the quotient polynomial, and the last number is the remainder.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing polynomials, particularly useful when dividing by linear factors of the form (x - c). It involves using the coefficients of the polynomial and performing a series of arithmetic operations to find the quotient and remainder without writing out the full polynomial long division.

Polynomial Coefficients

In a polynomial, coefficients are the numerical factors that multiply the variable terms. For example, in the polynomial 6x^5 - 2x^3 + 4x^2 - 3x + 1, the coefficients are 6, -2, 4, -3, and 1. Understanding how to identify and manipulate these coefficients is crucial for performing synthetic division.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder of this division is equal to f(c). This theorem is useful in synthetic division as it allows us to quickly find the remainder without fully performing the division, providing insight into the behavior of the polynomial at specific values.