Skill: Relate zeros to x-intercepts

A zero of a polynomial function is an input value that makes the function equal to zero:

$$f(c)=0.$$

On the graph, this means the point is $(c,0)$, which lies on the $x$-axis. Therefore, the real zeros of a polynomial are the $x$-coordinates of its $x$-intercepts.

Example

Find the real zeros and the $x$-intercepts of

$$f(x)=(x-2)(x+3)(x^2+4).$$

Step 1: Set the function equal to zero.

$$(x-2)(x+3)(x^2+4)=0$$

Step 2: Set each factor equal to zero.

For the linear factors:

$$x-2=0 \quad \Rightarrow \quad x=2$$ $$x+3=0 \quad \Rightarrow \quad x=-3$$

For the remaining factor:

$$x^2+4=0 \quad \Rightarrow \quad x^2=-4$$

This has no real solutions because the square of a real number cannot be negative. Its solutions are complex numbers, so they do not produce $x$-intercepts.

Step 3: State the real zeros.

The real zeros are

$$x=-3 \quad \text{and} \quad x=2.$$

Step 4: Write the $x$-intercepts.

Use each real zero as the $x$-coordinate and pair it with $0$:

$$\boxed{(-3,0)\text{ and }(2,0)}.$$

So, the real solutions of $f(x)=0$ are exactly the $x$-coordinates of the graph’s $x$-intercepts.