Zeros and Factors of Polynomials
Aligned to A-APR.B.3 — Common Core State Standards for Mathematics.
What this lesson teaches
The zeros of a polynomial are the x-values where it equals 0 — its factors reveal them, and they are where the graph crosses the x-axis.
Worked examples
Example 1
f(x) = x³ − x = x(x−1)(x+1). Zeros at x = 0, 1, −1, so the graph crosses the x-axis at those three points.
Example 2
f(x) = x³ − 4x² + 3x = x(x² − 4x + 3) = x(x − 1)(x − 3). Zeros at x = 0, 1, 3. Between consecutive zeros the sign of f can only change at a zero, so testing one point per interval — f(0.5) = 0.5(−0.5)(−2.5) = 0.625 > 0 — maps out the whole sign chart without plotting.
Example 3
Common mistake: reading a factor's sign backwards. The factor (x − 5) gives the zero x = +5, and (x + 5) gives x = −5 — you set each factor equal to ZERO and solve, rather than copying the number inside. Also, a repeated factor changes the picture: in f(x) = x(x − 2)², the graph TOUCHES the axis at x = 2 and turns back, instead of crossing, because the factor appears an even number of times.
Practice questions
- Find the zeros of (x−2)(x+5).
- Factor x² − 4 and state its zeros.
- How many real zeros can a cubic have at most?
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Ceros y Factores de Polinomios
Los ceros de un polinomio son los valores de x donde es igual a 0 — sus factores los revelan, y son los puntos donde la gráfica cruza el eje x.
Ejemplo 1: f(x) = x³ − x = x(x−1)(x+1). Ceros en x = 0, 1, −1, por lo que la gráfica cruza el eje x en esos tres puntos.
Ejemplo 2: f(x) = x³ − 4x² + 3x = x(x² − 4x + 3) = x(x − 1)(x − 3). Ceros en x = 0, 1, 3. Entre ceros consecutivos el signo de f solo puede cambiar en un cero, así que probar un punto por intervalo — f(0.5) = 0.5(−0.5)(−2.5) = 0.625 > 0 — traza toda la tabla de signos sin graficar.
Ejemplo 3: Error común: leer al revés el signo de un factor. El factor (x − 5) da el cero x = +5, y (x + 5) da x = −5 — se iguala cada factor a CERO y se resuelve, en vez de copiar el número de adentro. Además, un factor repetido cambia el dibujo: en f(x) = x(x − 2)², la gráfica TOCA el eje en x = 2 y regresa, en lugar de cruzarlo, porque el factor aparece un número par de veces.
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