Numbers & Functions · 4 Polynomials, Factoring, and the Quadratic Formula

4.3 Roots: where the machine hits zero

Here’s the single most useful question you can ask about a polynomial: For what inputs does the machine output exactly zero?

Those special inputs are called the roots (or zeros). On a graph, they’re exactly where the curve crosses or touches the horizontal axis. Why care so much? Because an astonishing amount of mathematics is secretly the question “when does this equal that?” — and you can always rearrange “this = that” into “this −that = 0.” So “when are two things equal?” becomes a question about roots. Finding roots is the engine under a huge fraction of all the algebra you’ll ever do.

4.4 Factoring: breaking a machine into pieces

Here’s a fact that looks small and turns out to be a superpower. If two things multiply to give zero, then at least one of them must have been zero. If A × B = 0, then A = 0 or B = 0 (or both). Now suppose I hand you a quadratic already written as a product: f(x) = (x −2)(x −3). When is it zero? Exactly when x −2 = 0 (so x = 2) or x −3 = 0 (so x = 3). The roots are 2 and 3, read straight off the pieces. Polynomials, Factoring, and the Quadratic Formula x f(x) (x −2) = 0 ⇒x = 2 (x −3) = 0 ⇒x = 3 f(x) = (x −2)(x −3) = x2 −5x + 6 Figure 4.1. Factoring as decomposition. Written as (x −2)(x −3), the curve’s roots fall out for free: it crosses zero exactly where each factor vanishes, at x = 2 and x = 3.

The catch is that polynomials usually arrive multiplied out: x2 −5x + 6. This is the same machine as (x −2)(x −3), but in this form the roots are hidden. Factoring is running that multiplication backwards. For x2 −5x + 6, factoring means finding two numbers that multiply to +6 and add to −5: those are −2 and −3, so it factors as (x −2)(x −3), and the roots are 2 and 3.

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