Divide using synthetic division. (x⁵+4x⁴−3x²+2x+3)÷(x−3)

Verified step by step guidance

1. Identify the divisor and rewrite it in the form \(x - c\). Here, the divisor is \(x - 3\), so \(c = 3\).
2. Write down the coefficients of the dividend polynomial \(x^{5} + 4x^{4} + 0x^{3} - 3x^{2} + 2x + 3\). Note that the \(x^{3}\) term is missing, so its coefficient is 0. The coefficients are: \([1, 4, 0, -3, 2, 3]\).
3. Set up the synthetic division by writing \(c = 3\) to the left and the coefficients in a row to the right.
4. Bring down the first coefficient (1) as it is. Then multiply it by \(c\) (3) and write the result under the next coefficient. Add the column and write the sum below. Repeat this multiply-and-add process for all coefficients.
5. After completing the process, the last number you get is the remainder. The other numbers form the coefficients of the quotient polynomial, which will be of degree one less than the original polynomial (degree 4 in this case).

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 and Missing Terms

When performing synthetic division, it is important to include coefficients for all powers of x, even if some terms are missing. For example, if the polynomial lacks an x³ term, its coefficient is zero, which must be included to maintain proper alignment during division.

Remainder and Quotient Interpretation

The result of synthetic division provides a quotient polynomial and a remainder. The quotient has one degree less than the original polynomial, and the remainder is a constant. Understanding how to interpret these results is essential for solving division problems and further factorization.