When looking for zeros of a rational function, why is the numerator equated to zero and not the denominator?

Question

If you have a function $F(x)=\dfrac{a(x)}{b(x)}$ and you are asked to find the zero(s) of the function, why do you set the numerator equal to zero, and not the denominator?

Answer 1

Because $a/b = 0$ if and only if $b\neq 0$ and $a=0$ .

If $b=0$, the expression $\frac ab$ is not defined.

Take a look at some examples.

• Is $\frac01=0$? Yes, because $0$ divided by $1$ is $0$.
• Is $\frac10=0$ ? No, because $\frac10$ is not a number.

Answer 2

The quotient

$$\frac{a}{b} = c$$

if there exists a unique number $c$ such that $a = bc$.

If $b = 0$, the quotient $a/b$ is undefined. To see this, consider cases.

Case 1: If $a \neq 0$ and $b = 0$, then

$$\frac{a}{b} = c \Rightarrow a = b \cdot c = 0 \cdot c = 0$$

contradicting our hypothesis that $a \neq 0$. Hence, if $a \neq 0$, the quotient does not exist.

Case 2: If $a = b = 0$, then $c = 0$ satisfies the equation $a = bc$ since $0 = 0 \cdot 0$. However, so does $c = 1$ since $0 = 0 \cdot 1$. Thus, the quotient is not uniquely defined.

Hence,

$$F(x) = \frac{a}{b}$$

is only defined if $b \neq 0$. If $F(x)$ is defined, then

$$F(x) = \frac{a}{b} = 0 \Rightarrow a = b \cdot 0 = 0$$

so $F(x)$ is only equal to zero if $a = 0$ and $b \neq 0$.