Relate zeros to x-intercepts

For a polynomial function, a real zero is an input value $c$ for which $f(c)=0$ ; on the graph, this produces the point $(c,0)$ , so the real solutions of $f(x)=0$ are exactly the $x$ -coordinates of the $x$ -intercepts. In factored form, each real linear factor identifies a zero and corresponding intercept, while complex zeros do not appear as $x$ -intercepts; detailed analysis of multiplicity and crossing behavior is not included.

Detailed Explanation: 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.

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Quadratic Discriminants - Equation to Graph

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