A coefficient is a number or other factor that multiplies a variable. For example, in the equation 3x – 4ky = 8, the 3 and 4k are coefficients of the variables x and y. In the expression ax – 2by3 + √3x2 - (a + 5)y there are four terms, and the coefficients are, in order, a, 2b, √3, and (a + 5). The multiplication sign between coefficient and literal expression is usualy omitted.
In the theory of equations, coefficients play an important part: thus in the cubic x3 + ax2 + bx + c = 0, it can be shown (in supposing the three roots are p, q, and r) that p + q + r = -a, pq + qr + rp = b, and pqr = –c.
The word coefficient combines three elements, the Latin facere ("to do"), and the prefixes ex ("out") and co ("with"), to give the overall meaning of joining two things together to bring about a result. The 16th century mathematician Francois Vieta may have coined word, but it was not commonly used until around the beginning of the 18th century.