Use synthetic division to perform each division. (3x^3+6x^2-8x+3)/(x+3)

Verified step by step guidance

1. Identify the coefficients of the dividend polynomial \(3x^3 + 6x^2 - 8x + 3\), which are \([3, 6, -8, 3]\).
2. Set up the synthetic division by writing the zero of the divisor \(x + 3\), which is \(-3\), to the left of the vertical bar.
3. Bring down the leading coefficient \(3\) to the bottom row.
4. Multiply \(-3\) by the number just written on the bottom row \(3\), and write the result \(-9\) under the next coefficient \(6\).
5. Add \(6\) and \(-9\) to get \(-3\), and write this result in the bottom row. Repeat the multiplication and addition process for the remaining coefficients.

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial division that allows for quicker calculations when dividing by a linear factor. It involves using the coefficients of the polynomial and a specific value derived from the divisor. This method is particularly useful for dividing polynomials of higher degrees, as it reduces the complexity of long division.

Polynomial Coefficients

In a polynomial, coefficients are the numerical factors that multiply the variable terms. For example, in the polynomial 3x^3 + 6x^2 - 8x + 3, the coefficients are 3, 6, -8, and 3, corresponding to the terms x^3, x^2, x, and the constant term, respectively. Understanding coefficients is essential for performing operations like synthetic division.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is equal to f(c). This theorem is useful in synthetic division as it helps to quickly determine the remainder without performing the full division process, providing insight into the behavior of the polynomial at specific values.