Solving the Difference of Polynomials: A Step-by-Step Guide

Polynomials are versatile mathematical expressions that represent a range of real-world phenomena. They crop up in algebraic equations, geometric shapes, and physical models. One of the common operations you’ll encounter when working with polynomials is their subtraction. The difference of polynomials involves subtracting one polynomial expression from another. In this post, we’ll provide a step-by-step guide to solving the difference of polynomials.

Before proceeding, let’s define what we mean by a polynomial. A polynomial is an expression that contains variables raised to non-negative integer powers and multiplied by coefficients. For instance, x^2 – 3x + 5 and 3y^3 + 2y – 1 are examples of polynomials. Poly means many, and nomial means terms, so polynomials consist of many terms.

To solve the difference of polynomials, follow these steps:

Step 1: Identify the two polynomials you want to subtract. Suppose you have P(x) = 2x^2 + 3x – 4 and Q(x) = x^2 – 2x + 1. You want to find the difference of P(x) and Q(x), which is P(x) – Q(x).

Step 2: Distribute the minus sign across the second polynomial. To subtract a polynomial, you need to add its additive inverse. In this case, you can distribute the minus sign across the terms of the second polynomial and make them negative. So, Q(x) becomes -x^2 + 2x – 1.

Step 3: Combine like terms. You can now subtract the two polynomials term by term in descending order of powers. Start from the highest degree term (in this case, x^2) and move to the lowest. So, P(x) – Q(x) = (2x^2 – x^2) + (3x – 2x) + (-4 – 1). Combine like terms in each bracket to get P(x) – Q(x) = x^2 + x – 5.

Step 4: Simplify the expression. You can further simplify the answer by factoring or grouping like terms. In this case, the expression is already simplified. So, P(x) – Q(x) = x^2 + x – 5 is the final answer.

That’s it! You’ve successfully solved the difference of polynomials. Remember to follow the order of operations, distribute the minus sign, and combine like terms. With practice, you’ll be able to handle more complex polynomials and solve them with ease.