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Polynomial division involves dividing a polynomial by another polynomial. In our example, we divided a cubic polynomial, \(x^3 - 4x^2 + 5x - 6\), by a binomial, \(x-3\). There are two common methods for polynomial division: long division and synthetic division. Synthetic division is often easier and quicker, particularly when the divisor is a binomial of the form \(x - c\). In synthetic division:

• We focus on the coefficients of the polynomial
• We use a streamlined process involving multiplication and addition

First, we set up the coefficients and the value derived from the divisor. Then, we perform the division step-by-step by multiplying and adding the results. Finally, we interpret the bottom row of numbers to find the coefficients of the quotient and any remainder.

Intermediate Algebra focuses on polynomials, rational expressions, radicals, and quadratic equations. Synthetic division is a crucial tool in this area, simplifying what could otherwise be a tedious process. Let's break down our step-by-step solution: We synthesize the polynomial's coefficients \( [1, -4, 5, -6] \). Next, using the constant term from the divisor \(x - 3\), which is 3, we carry out synthetic division by:

• Bringing down the first coefficient
• Multiplying and adding across each coefficient

This method ensures accuracy and helps to build a strong foundation in manipulating and understanding polynomial expressions. These skills are essential for solving complex algebraic problems, as well as for calculus and other advanced math courses.

Coefficients are the numerical values in front of the variables in a polynomial. In the polynomial \(x^3 - 4x^2 + 5x - 6\), the coefficients are 1, -4, 5, and -6. Properly identifying and manipulating coefficients are critical in synthetic division. Here’s how we use them:

• Write the coefficients in order initially: \([1, -4, 5, -6]\)
• Perform calculations by combining multiplication and addition

For instance, the multiplication step involves the divisor's constant (3 in \(x-3\)). We multiply this constant with the coefficient values and sum them up through each step. This method helps simplify complex division into more manageable steps, making it easier to interpret and understand the results. Knowing how to work with coefficients builds the essential algebraic skills to tackle polynomial equations effectively.