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

Verified step by step guidance

1. Identify the divisor and the dividend. Here, the divisor is \(x - 3\) and the dividend is \(x^5 + 4x^4 - 3x^2 + 2x + 3\).
2. Set up the synthetic division. Write the coefficients of the dividend: \([1, 4, 0, -3, 2, 3]\). Note the zero for the missing \(x^3\) term.
3. Use the root of the divisor, which is \(3\), as the number to the left of the synthetic division setup.
4. Bring down the leading coefficient (1) to the bottom row.
5. Multiply the number you just brought down by the root (3) and write the result under the next coefficient. Add this result to the next coefficient and write the sum below the line. Repeat this process for each coefficient.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing polynomials, particularly useful when dividing by linear factors. It involves using the coefficients of the polynomial and a specific value (the root of the divisor) to perform the division in a more efficient manner than traditional long division. This technique streamlines calculations and is especially advantageous for higher-degree polynomials.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The degree of the polynomial is determined by the highest power of the variable. Understanding polynomial functions is crucial for performing operations like addition, subtraction, multiplication, and division, as well as for analyzing their behavior and graphing.

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 provides a quick way to evaluate the remainder without performing full polynomial division, and it is particularly useful in synthetic division, where the value of c is used directly in the calculations.