What are the rules for combining polynomials?

Combining polynomials involves simplifying expressions by adding or subtracting them. The core rules ensure that you correctly group and manipulate like terms. Here's a breakdown of the key rules:

Steps to Combine Polynomials

1. Remove the Brackets: This is the first step to simplify the expression (Rule 1). Pay close attention to signs, especially if there's a negative sign in front of a bracket. For example, if you have -(x + 2), it becomes -x - 2 when the bracket is removed.

2. Organize Terms (Descending Order): Arrange the terms in descending order based on their degree (exponent of the variable). The constant terms (numbers without variables) should be placed at the end (Rule 2). For instance, 3x + 5x^2 + 2 would be re-organized as 5x^2 + 3x + 2.

3. Combine Like Terms: Identify and group "like terms," which are terms that have the same variable raised to the same power (Rule 3). For example, in the expression 4x^2 + 2x + x^2 - x, 4x^2 and x^2 are like terms, as are 2x and -x.

4. Add Coefficients: Add (or subtract) the coefficients (the numbers in front of the variables) of the like terms, keeping the variable and exponent the same (Rule 4). So, 4x^2 + x^2 becomes 5x^2, and 2x - x becomes x.

Example

Let's say you want to combine the following polynomials:

(3x^2 + 2x - 1) + (x^2 - x + 4)

Here's how to apply the rules:

1. Remove the Brackets: 3x^2 + 2x - 1 + x^2 - x + 4

2. Organize Terms (Descending Order): Already organized in this example.

3. Combine Like Terms: (3x^2 + x^2) + (2x - x) + (-1 + 4)

4. Add Coefficients: 4x^2 + x + 3

Therefore, the simplified expression is 4x^2 + x + 3.