Coefficient

Algebra-1

1. Fundamental Concepts

A Coefficient refers to the numerical factor (including its sign) that multiplies the variable part in a term.
- Core Feature: It is the "numerical and symbolic prefix" of the variable part in a term; if a term has no explicit numerical factor, its coefficient is 1 or -1 (not 0).
- Special Case: For a constant term (a term without variables, e.g., "5" or "-3"), its coefficient is the constant itself.

2. Key Concepts

Key Concept	Definition	Example
Explicit Coefficient	A coefficient clearly written in the term (with a visible number and sign).	In "-7xy", the coefficient is -7; in "3x²", it is 3.
Implicit Coefficient	A coefficient not written explicitly (default 1 for positive terms, -1 for negative terms).	In "x" (same as "1×x"), coefficient is 1; in "-ab" (same as "-1×ab"), it is -1.

3. Examples

(1) Easy

Example 1: What is the coefficient of "-ab" in "3a²b - ab + 4b²"?
Solution: "-ab" has no explicit number, but a negative term without a visible number has an implicit coefficient of -1 (since "-ab" = "-1×ab"). Answer: -1.

(2) Medium

Example 1: Find the coefficient of "x²y" in "5x²yz - 3x²y + 2z".
Solution: In "-3x²yz", factors other than "x²y" are "-3". Answer: -3.

(3) Difficult

Example 1: Simplify "2x - 3(x + y)" and find the coefficient of "x".
Solution: Simplify first: \(2x - 3x - 3y = -x - 3y\); coefficient of "x" is -1.
Example 2: In "\((4a²b - 2ab) + 3ab\)", find the coefficient of "ab" after combining like terms.
Solution: Combine like terms: \(4a²b + (-2ab + 3ab) = 4a²b + ab\); coefficient of "ab" is 1.

4. Problem-Solving Techniques

Technique 1: Identify Coefficients (2-Step Check)

Distinguish variable vs. numerical parts: Separate the variable part (letters and their powers) from the rest.
Check for implicit coefficients: If no number precedes the variable, use 1 (positive term) or -1 (negative term).

Technique 2: Coefficients After Simplification

First combine like terms (simplify the expression via removing parentheses, merging similar variable parts), then identify the coefficient of the target term.