Geometrical meaning of zeroes

So far we have called $k$ a zero of $p(x)$ if $p(k) = 0$ . There is a beautiful picture of this in the $xy$ -plane.

Idea: graphs and the $x$ -axis

For each polynomial $p(x)$ we draw the graph $y = p(x)$ in the coordinate plane. A point $(k, 0)$ on this graph means $p(k) = 0$ . So the zeroes of $p(x)$ are exactly the $x$ -coordinates of the points where the graph crosses (or touches) the $x$ -axis.

This visual gives us a powerful way to count zeroes without doing algebra: just count how many times the graph hits the $x$ -axis.

Theorem / Concept: shapes by degree

Linear. The graph of $y = ax + b$ ( $a \ne 0$ ) is a straight line with slope $a$ . It crosses the $x$ -axis at exactly one point: $x = -b/a$ . So a linear polynomial has exactly one zero.

Quadratic. The graph of $y = ax^2 + bx + c$ ( $a \ne 0$ ) is a parabola, opening upwards if $a > 0$ and downwards if $a < 0$ . The parabola can:

cross the $x$ -axis at two distinct points , the quadratic has two distinct real zeroes;
touch the $x$ -axis at exactly one point (vertex on the axis) , the quadratic has one repeated (double) zero;
not meet the $x$ -axis at all , the quadratic has no real zeroes.

Which case occurs is decided by the discriminant $D = b^2 - 4ac$ :

$D > 0$ : two distinct real zeroes.
$D = 0$ : one repeated real zero (at the vertex).
$D < 0$ : no real zeroes (the parabola sits entirely above or entirely below the axis).

Cubic. The graph of $y = ax^3 + bx^2 + cx + d$ ( $a \ne 0$ ) is an "S"-shaped curve that rises from $-\infty$ to $+\infty$ (if $a > 0$ ) or vice versa. It must cross the $x$ -axis at least once, and at most three times. So a cubic has $1$ , $2$ (with one repeated zero), or $3$ real zeroes.

In general, a polynomial of degree $n$ meets the $x$ -axis at most $n$ times , its graph cannot wiggle more than that.

This geometric view is wonderful for reading off the structure of a polynomial from its sketch. If a board exam paper shows a parabola crossing the $x$ -axis at $-1$ and $4$ , you instantly know the zeroes , even if no equation is given.

Worked examples

Example 1. A parabola has its vertex on the $x$ -axis at $(3, 0)$ . How many zeroes does the corresponding quadratic have? Name them.

Touching the $x$ -axis at one point means one repeated (double) zero. The zero is $x = 3$ (with multiplicity $2$ ).

Example 2. A cubic crosses the $x$ -axis at $x = -2$ , touches it at $x = 1$ , and the graph never meets the axis again. How many zeroes does it have?

Touching counts as a repeated zero. So zeroes are $-2$ , $1$ , $1$ , that is, three zeroes counting multiplicity, with two distinct values. Many textbooks (and our chapter) count them by distinct values; here that count is $2$ .

Example 3. The graph of a quadratic lies entirely above the $x$ -axis. What can you say about its discriminant?

No intersection with the $x$ -axis means no real zeroes, so $D < 0$ . Also the leading coefficient $a > 0$ (parabola opens upward).

Example 4. Sketch (qualitatively) $y = x^2 - 4$ . Read its zeroes from the picture.

The parabola opens upward, has vertex at $(0, -4)$ , and crosses the $x$ -axis at $x = \pm 2$ . Zeroes: $-2$ and $2$ .

Example 5. A polynomial has degree $4$ and its graph crosses the $x$ -axis at $3$ distinct points. How many real zeroes does it have? Could it have $5$ ?

A degree- $4$ polynomial has at most $4$ real zeroes (counted with multiplicity). It cannot have $5$ . Three crossings means at least three distinct real zeroes; a possible fourth zero is hiding either as a repeated touch or as a non-real (complex) zero.

Try it yourself

The graph of a quadratic touches the $x$ -axis at $(2, 0)$ and nowhere else. State the zeroes.
The graph of $y = x^2 + 1$ , how many real zeroes does it have? Why?
A cubic has zeroes $-2, 0, 3$ . Sketch its graph (sign at each region).
From a sketch, the parabola opens downward and crosses the $x$ -axis at $-1$ and $5$ . Write a possible quadratic.
Can the graph of a quartic ( $\deg = 4$ ) cross the $x$ -axis $5$ times? Justify.
The discriminant of a quadratic is $0$ . Describe its graph.
A cubic with positive leading coefficient has only one real zero. Sketch its graph.
The graph of $p(x) = x^3 - 3x + 2$ touches the $x$ -axis once and crosses it once. Find the zeroes.
Argue that every cubic with real coefficients has at least one real zero.
From the graph of $y = p(x)$ , the curve never meets the $x$ -axis. What does that say about its zeroes?

Pitfalls / Insight

"No real zeroes" is allowed for quadratics, but never for cubics. Cubics always cross the $x$ -axis at least once.
A graph touching the $x$ -axis (tangent) means a repeated zero, not just one.
Sign of leading coefficient controls end behaviour. Positive $\Rightarrow$ curve goes to $+\infty$ on the right.

Insight. Algebra ( $p(x) = 0$ ) and geometry (graph meets the $x$ -axis) are two faces of the same idea. Switching between them is the single most useful skill of this chapter.