Mathématiques

What Is a Coefficient?

In 5x² − x + 7 the coefficients are 5 and −1, with a hidden 0 where a term is missing. Why the sign belongs to the coefficient, and what the SAT calls 'a'.

Coefficient

A coefficient is the number multiplying a variable in a term, such as the 5 in 5x².

Où les élèves le rencontrent: Grade 7 mathematics when the parts of a term are named, and constantly afterwards at GCSE, IGCSE and on the Digital SAT, which asks directly for the value of a coefficient.

La réponse en bref
A coefficient is the number multiplying a variable in a term. In 5x² − x + 7 the coefficient of x² is 5 and the coefficient of x is −1, because the invisible 1 carries the minus sign with it. The 7 is a constant term, not a coefficient.

Un exemple

5x² − x + 7

Coefficient of x²: 5. Coefficient of x: −1, since −x means −1 × x. Constant term: 7. The 7 has no variable attached, so it is not a coefficient of anything. If the expression were 5x² + 7, the coefficient of x would be 0 — the term is not missing so much as present with nothing in front of it.

Reading the coefficients off an expression

Split the expression into terms at the + and − signs, then look at each term for a number and a variable. The number is the coefficient of whatever variable that term contains, including its power.

Two cases are worth naming because they are invisible. A term written as x or −x has a coefficient of 1 or −1, since nobody writes 1x. And a power that does not appear at all has a coefficient of 0: in x² + 9 the coefficient of x is zero, which matters the moment you put that quadratic into the formula with b = 0.

The sign belongs to the coefficient

In 3x² − 7x + 2, the coefficients are a = 3, b = −7 and c = 2. The minus sign is part of b, not a separate thing sitting between two terms, and once b = −7 goes into the quadratic formula, −b becomes +7 and b² becomes 49.

Writing b = 7 here is one of the most reliable ways to lose a quadratic. It produces a discriminant that is often right by luck, since squaring hides the error, and then a completely wrong pair of roots from the −b at the front.

Where the SAT uses the word

The Digital SAT's Algebra and Advanced Math questions frequently give an expression or equation with a letter standing in for a coefficient and ask what that letter must be. The phrasing is usually 'what is the value of a' rather than 'what is the coefficient', but the object is the same.

The vertex form of a quadratic, y = a(x − h)² + k, is where this comes up most often: a is the coefficient that controls the shape, and h and k position the curve. Questions that give two equivalent forms of one quadratic and ask for a coefficient are really asking you to expand or complete the square and then compare the numbers in front of matching powers.

Questions fréquentes

Is 7 a coefficient in 5x² − x + 7?

No. It is the constant term. A coefficient has to be multiplying a variable, and 7 stands alone. Some texts loosely describe it as 'the coefficient of x⁰', since x⁰ = 1, but on a school or SAT question you should call 7 the constant and reserve 'coefficient' for 5 and −1.

What is the coefficient of x in x² + 9?

Zero. The x term is absent, which is the same as being there with a coefficient of 0. This is not a technicality: solving x² + 9 = 0 with the quadratic formula requires a = 1, b = 0 and c = 9, and treating b as though it had no value at all stalls the calculation.

Is the coefficient of −x equal to 1 or −1?

−1. The term −x is shorthand for −1 × x, so the coefficient carries the minus sign. Anywhere you need to collect terms, substitute, or read values into a formula, use −1. Reading it as 1 flips the sign of that term and everything downstream of it.

What is a leading coefficient?

The coefficient of the highest power in the expression, once it is written in descending order of powers. In 5x² − x + 7 the leading coefficient is 5. It decides a great deal about a graph: for a quadratic, a positive leading coefficient gives a curve that opens upwards and a negative one opens downwards.

Can a coefficient be a fraction or a letter?

Both. In ½n² + ½n the coefficients are fractions, and in ax² + bx + c the coefficients are the letters a, b and c standing in for numbers we have not been told. A coefficient just has to be the multiplying part of a term; it does not have to be a whole number.

Sources

SAT Suite of Assessments: The Math Section — College Board

Dernière mise à jour 21 août 2026