Math101 · Independent Ontario learning library
Math101learn.math101.caCoefficient
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.
Worked example
Representations and interpretation
Algebra tiles represent a coefficient as the signed count of identical variable tiles: $3x$ uses three positive $x$-tiles, while $-2x$ uses two negative $x$-tiles. In a table, coefficient and variable part form separate columns.
Reasoning about variations
In $5(a+b)$, $5$ multiplies the entire group; after distribution, it becomes the coefficient of both $a$ and $b$. In $kx$, the coefficient of $x$ is $k$, showing that coefficients need not be fixed numerals.
Common mistakes
How to check your work
- Multiply the stated coefficient by the variable part and recover the exact term.
- Substitute $1$ for all variables; the term’s value should equal its coefficient.
- Compare only like variable parts before combining coefficients.
Practice
- What is the coefficient of $x^2y$ in $-12x^2y$?
- What is the coefficient of $m$ in $m-7$?
- In $3ab+4a$, are the terms like terms?
Answers and brief solutions
Show answers
- $-12$ $-12$ is the signed numerical factor multiplying $x^2y$.
- $1$ $m=1m$.
- No Their variable parts, $ab$ and $a$, differ.
Synthesis and transfer
In a cost model, the coefficient of quantity represents a per-item rate; changing that coefficient alters the slope without changing the fixed starting cost.
If the model is $C=4.75n+12$, the coefficient $4.75$ carries units of dollars per item and determines how much total cost changes when $n$ increases by one. The constant $12$ remains when no items are purchased, so the two numbers have different contextual jobs. A negative coefficient would describe a quantity that decreases as the input grows, while a zero coefficient removes variable dependence from that term. In a polynomial such as $-x+6x^2$, the first coefficient is $-1$, not absent, and the terms cannot be combined because their powers differ. Reading coefficients with their attached variable parts makes both algebraic simplification and model interpretation more precise.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the coefficient of $x^2y$ in $-12x^2y$?
- $-12$ is the signed numerical factor multiplying $x^2y$.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
