Solving Polynomial Equations by Graphing

Q: Which intercepts are the solutions?

The $x$-intercepts. They are where $y = 0$, which is exactly the equation $p(x) = 0$.

Q: Is the $y$-intercept ever a solution?

Not of $p(x) = 0$. It is the value $p(0)$, the constant term — only a solution if it happens to lie on the x-axis.

Q: Can graphing find complex roots?

No. Only real roots show as $x$-intercepts. Complex roots must be found algebraically.

Just like quadratics, the real solutions of any polynomial equation $p(x) = 0$ are the $x$ -intercepts of its graph — where the curve crosses the x-axis.

Graphing shows the real roots at a glance. The $y$ -intercept is a distraction here: it is where the graph meets the y-axis, not a solution of $p(x) = 0$ .

Real roots are $x$ -intercepts

Solving $p(x) = 0$ means finding where $y = 0$ — the x-axis. Each crossing is a real solution.

If the graph crosses at $x = -4$ and $x = 1$ , those are the real solutions, regardless of where it crosses the y-axis.

In the graph below, the two places the curve meets the x-axis are the real solutions of $p(x) = 0$ ; where it meets the y-axis is the constant term, not a solution.

Don't confuse the $y$ -intercept

The $y$ -intercept ( $x = 0$ ) tells you the constant term, not a root. It is never a solution of $p(x) = 0$ unless it happens to sit on the x-axis too.

Focus on x-axis crossings. Complex roots do not appear on the real graph at all.

Worked examples

Example 1: reading real roots

A graph of $y = p(x)$ crosses the x-axis at $x = -4$ and $x = 1$ , and the y-axis at $y = 3$ . What are the real solutions of $p(x) = 0$ ?

Solutions are

x

-intercepts only

x = -4, ; x = 1

The

y

-intercept is not a solution

\text{ignore } y = 3

Answer: $x = -4$ and $x = 1$

Example 2: what the $y$ -intercept means

What does the $y$ -intercept $y = 3$ tell you?

It is p(0)

p(0) = 3

The constant term, not a root

\text{not a solution}

Answer: The constant term $p(0) = 3$

Example 3: fewer crossings than the degree

The graph of a degree-3 polynomial $p(x)$ crosses the x-axis only at $x = 2$ . How many real solutions does $p(x) = 0$ have?

One crossing means one real solution

x = 2

The other two roots are complex, so they never meet the axis

\text{not visible on the graph}

Answer: One real solution, $x = 2$

Try one yourself

Common questions

Which intercepts are the solutions?

The $x$ -intercepts. They are where $y = 0$ , which is exactly the equation $p(x) = 0$ .

Is the

y

-intercept ever a solution?

Not of $p(x) = 0$ . It is the value $p(0)$ , the constant term — only a solution if it happens to lie on the x-axis.

Can graphing find complex roots?

No. Only real roots show as $x$ -intercepts. Complex roots must be found algebraically.

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Solving Polynomial Equations by Graphing

Real roots are $x$ -intercepts

Don't confuse the $y$ -intercept

Worked examples

Example 1: reading real roots

Example 2: what the $y$ -intercept means

Example 3: fewer crossings than the degree

Try one yourself

Common questions

Keep learning

Solving Polynomial Equations Algebraically

Roots & Zeros

Analyzing Graphs of Polynomial Functions

Solving Polynomial Equations by Graphing

Real roots are xxx-intercepts

Don't confuse the yyy-intercept

Worked examples

Example 1: reading real roots

Example 2: what the yyy-intercept means

Example 3: fewer crossings than the degree

Try one yourself

Common questions

Keep learning

Solving Polynomial Equations Algebraically

Roots & Zeros

Analyzing Graphs of Polynomial Functions

Real roots are $x$ -intercepts

Don't confuse the $y$ -intercept

Example 2: what the $y$ -intercept means