Lesson 5: Solving Polynomial Equations

Property

Zero of a Function
For any function $f$, if $f(x) = 0$, then $x$ is a zero of the function.

When we find the values of $x$ for which $f(x) = 0$, we are finding the zeros of the function. When $f(x) = 0$, the point $(x, 0)$ is a point on the graph, which is an $x$-intercept.

Examples

• For the function $f(x) = x^2 - x - 8$, find $x$ when $f(x) = 4$. Set $x^2 - x - 8 = 4$, which simplifies to $x^2 - x - 12 = 0$. Factoring gives $(x - 4)(x + 3) = 0$, so $x = 4$ and $x = -3$.

Property

For a polynomial equation $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0$ with integer coefficients, any rational solution $\frac{p}{q}$ (in lowest terms) must satisfy:

Property

If a polynomial has rational coefficients and $a + b\sqrt{c}$ is a zero (where $a$ and $b$ are rational and $\sqrt{c}$ is irrational), then $a - b\sqrt{c}$ must also be a zero.

Examples