How the zeroes of a polynomial connect to its coefficients
A polynomial in x is an expression made of terms like axⁿ, where n is a whole number. Its degree is the highest power of x: linear polynomials have degree 1, quadratics degree 2 and cubics degree 3. A zero of a polynomial p(x) is a value of x that makes p(x) = 0. On a graph, the zeroes are the x-coordinates where the curve y = p(x) meets the x-axis, so a quadratic can have at most two zeroes and a cubic at most three.
For a quadratic ax² + bx + c with zeroes α and β, the sum α + β = −b/a and the product αβ = c/a. This lets you build a quadratic from its zeroes as x² − (sum)x + (product). For a cubic ax³ + bx² + cx + d, the sum of the zeroes is −b/a and their product is −d/a. The division algorithm states that p(x) = g(x) × q(x) + r(x), where the remainder r(x) has a smaller degree than the divisor g(x). By the factor theorem, (x − a) is a factor of p(x) exactly when p(a) = 0, which is the usual way to begin factorising a cubic.
Why it matters
Polynomials are the building blocks of algebra and appear in almost every later chapter, including quadratic equations, coordinate geometry and calculus. Engineers and economists use polynomial curves to model the path of a ball, the shape of an arch or how cost changes with output. Knowing the zero–coefficient relationships saves time in board questions that ask you to find a polynomial or an unknown coefficient.
Key terms
- Degree
- The highest power of the variable that appears in a polynomial.
- Zero of a polynomial
- A value of the variable for which the polynomial equals zero.
- Factor theorem
- The result that (x − a) divides p(x) exactly if and only if p(a) = 0.
- Division algorithm
- The identity dividend = divisor × quotient + remainder, applied to polynomials.