Use synthetic division to perform each division. (3x³+6x²-8x+3)/(x+3)

Verified step by step guidance

1. Identify the divisor and rewrite it in the form \(x - c\). Since the divisor is \(x + 3\), rewrite it as \(x - (-3)\), so \(c = -3\).
2. Write down the coefficients of the dividend polynomial \(3x^3 + 6x^2 - 8x + 3\). These are \(3\), \(6\), \(-8\), and \(3\).
3. Set up the synthetic division by placing \(c = -3\) to the left and the coefficients in a row: \(3 \quad 6 \quad -8 \quad 3\).
4. Bring down the first coefficient \(3\) as it is. Then multiply it by \(c = -3\) and write the result under the next coefficient. Add the column and repeat this process for all coefficients.
5. The numbers you get at the bottom row (except the last one) are the coefficients of the quotient polynomial, and the last number is the remainder. Write the quotient in descending powers of \(x\) starting from one degree less than the original polynomial.

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, making calculations faster and less error-prone.

Polynomial Coefficients

Polynomial coefficients are the numerical factors in front of the variable terms. In synthetic division, these coefficients are arranged in descending order of degree and manipulated to find the quotient and remainder efficiently.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by x - c, the remainder is f(c). This concept helps verify the result of synthetic division by evaluating the polynomial at the root of the divisor.