Math101 · Independent Ontario learning library
Math101Coefficient
A coefficient is the numerical factor multiplying a variable or variable product in a term. In $-7x^2y$, the coefficient is $-7$; in $x$, the understood coefficient is $1$; in $-x$, it is $-1$.
Coefficients encode scale, direction, rate, and weight. Recognizing them correctly is essential for collecting like terms, graphing linear equations, reading polynomials, and interpreting formulas.
Intuition and core definition
A coefficient is the numerical factor multiplying a variable or variable product in a term. In $-7x^2y$, the coefficient is $-7$; in $x$, the understood coefficient is $1$; in $-x$, it is $-1$. A coefficient includes its sign and does not include the variable part.
Notation, language, and conditions
A term can be viewed as coefficient times variable part: $ax^n$. In $ax^n$, $a$ is a coefficient, $x$ a variable, and $n$ an exponent. In an expression such as $3x+5$, $5$ is a constant term, not the coefficient of $x$. Parameters such as $m$ in $y=mx+b$ act as coefficients of variables even if their numerical values are not yet known.
Why this idea matters
A coefficient records how strongly a variable term is scaled, including the implied values $1$ and $-1$ that are often not printed.
A dependable method
- Separate an expression into terms at addition or subtraction signs, keeping each sign attached.
- Within the target term, identify all variable factors and exponents.
- The remaining numerical factor, including its sign, is the coefficient.
- If no number is written, use an understood coefficient of $1$ or $-1$.
- Reconstruct coefficient times variable part to check the original term.
