Coefficient
Pre-AlgebraA coefficient is the numerical factor multiplied by the variable in an algebraic term, such as 5 in the term 5x.
Definition
A coefficient is the number written right in front of a variable. It tells you how many of that variable you have.
Example
In $7x$, the coefficient is $7$. That means $7$ groups of $x$. In $3y + 2$, the coefficient of $y$ is $3$.
Key Insight
If you see just $x$ with no number in front, the coefficient is secretly $1$, because $1$ times anything is itself.
Definition
A coefficient is the numerical factor of a variable term in an algebraic expression. It multiplies the variable part. A term like $x$ has an implied coefficient of $1$, and $-x$ has an implied coefficient of $-1$.
Example
In $6x^2 - 4x + 9$, the coefficient of $x^2$ is $6$, and the coefficient of $x$ is $-4$. The $9$ is the constant term, not a coefficient.
Key Insight
Coefficients determine the "weight" of each variable term. When combining like terms, you add or subtract coefficients while keeping the variable part unchanged.
Definition
In a polynomial $p(x) = a_nx^n + \ldots + a_1x + a_0$, each $a_i$ is the coefficient of the corresponding power of $x$. Coefficients belong to a ring or field (e.g., the integers, rationals, or reals). In linear algebra, coefficients appear in linear combinations: $v = c_1v_1 + c_2v_2 + \ldots + c_nv_n$.
Example
In the polynomial $3x^3 - x^2 + 5$, the leading coefficient is $3$ (for $x^3$), the coefficient of $x^2$ is $-1$, the coefficient of $x$ is $0$, and the constant term is $5$.
Key Insight
The leading coefficient of a polynomial determines end behavior and scaling; it plays a central role in factoring, the Rational Root Theorem, and polynomial division.