Coefficient

Notation

In the expression

$$3x + 7$$

the coefficient of $x$ is $3$. The $7$ has no variable attached, so it is a constant, not a coefficient.

The sign is part of the coefficient

This is the detail that causes the most lost marks. In

$$-5y + 2$$

the coefficient of $y$ is $-5$, not $5$. When you divide both sides of an equation by the coefficient, you divide by $-5$, sign included — and in an inequality that negative is exactly what flips the sign.

When you cannot see a coefficient

If a variable appears with no number in front, its coefficient is $1$:

$$x = 1x \qquad {-x} = -1x$$

Writing the invisible $1$ in helps when you are collecting like terms, because $x + 3x$ is $1x + 3x = 4x$ rather than $3x$.

Leading coefficient

In a quadratic written in standard form $ax^2 + bx + c$, the number $a$ is called the leading coefficient — the coefficient of the highest power. It decides whether a parabola opens upward or downward, and it must be divided out before completing the square.