Geometrical Meaning of the Zeroes of a Polynomial

Zeroes on the Graph — The Big Idea

In the last section a zero of $p(x)$ was a number $k$ with $p(k) = 0$. Now let's see what that means on a graph.

When we plot $y = p(x)$, we get a curve. The zeroes of $p(x)$ are exactly the x-coordinates of the points where the graph cuts (or touches) the x-axis. This is because on the x-axis, $y = 0$, and $y = p(x) = 0$ is precisely the condition for a zero.

Key Point: Geometrically, the zeroes of a polynomial are the x-intercepts of its graph — the points where $y = 0$.

Think of it this way: solving $p(x) = 0$ algebraically and finding where the graph crosses the x-axis are two views of the same thing.

[Board Important] A common 1-mark question shows a graph and asks for the number of zeroes. Just count how many times the curve meets the x-axis.

Linear Polynomial — A Straight Line

The graph of a linear polynomial $y = ax + b$ is a straight line.

• A straight line (that is not horizontal) crosses the x-axis at exactly one point.
• That point is $\left(-\dfrac{b}{a},\ 0\right)$, so a linear polynomial has exactly one zero.

Example

For $y = 2x - 4$, the line meets the x-axis where $2x - 4 = 0$, i.e. at $x = 2$. So the single zero is 2, matching the one x-intercept.

Key Point: A non-constant linear polynomial always has exactly one real zero — its graph crosses the x-axis once.

[Board Important] A horizontal line like $y = 3$ (a non-zero constant) never meets the x-axis, so it has no zero — consistent with a degree-0 polynomial.

Quadratic Polynomial — A Parabola

The graph of a quadratic $y = ax^2 + bx + c$ is a U-shaped curve called a parabola.

• If $a > 0$, the parabola opens upwards (∪).
• If $a < 0$, it opens downwards (∩).

The number of zeroes depends on how the parabola meets the x-axis:

Situation	Zeroes
Cuts the x-axis at two distinct points	2 distinct zeroes
Just touches the x-axis at one point	1 zero (repeated/equal)
Does not meet the x-axis	0 real zeroes

Key Point: A quadratic has at most 2 real zeroes. 'Touching' the axis means the two zeroes coincide (equal roots).

[Board Important] When a parabola only touches the x-axis, the polynomial has two equal zeroes — examiners count this as one distinct zero but two coincident roots. Read the question's wording carefully.

Cubic Polynomial and the General Rule

The graph of a cubic $y = ax^3 + bx^2 + cx + d$ is a smooth curve that can bend twice.

• A cubic can meet the x-axis at 3, 2, or 1 point(s), so it has at most 3 real zeroes.
• It always meets the x-axis at least once (so a cubic always has at least one real zero).

The general principle

For any polynomial: the number of real zeroes is at most equal to its degree, because that is the maximum number of times its graph can cross the x-axis.

Key Point: Degree $n$ ⇒ at most $n$ real zeroes ⇒ graph meets the x-axis at most $n$ times.

[Board Important] If a graph is shown crossing the x-axis at 3 points, the minimum possible degree of the polynomial is 3. 'Number of x-intercepts ≤ degree' is the key reasoning.

Solved Examples

Example 1: Zeroes from a graph (count)

A graph cuts the x-axis at exactly 2 points. How many zeroes does the polynomial have?

Solution:

1. Each x-intercept corresponds to one zero.
2. Two intercepts ⇒ two zeroes.

Final Answer: 2 zeroes.

Takeaway: Number of zeroes = number of times the graph meets the x-axis.

Example 2: Linear graph zero

Where does the graph of $y = 2x - 4$ cross the x-axis, and what is the zero?

Solution:

1. On the x-axis $y = 0$: $2x - 4 = 0$.
2. $x = 2$, so the graph crosses at $(2, 0)$.

Final Answer: Crosses at $(2,0)$; the zero is 2.

Takeaway: Set $y = 0$ to find the x-intercept = the zero.

Example 3: Parabola opening direction

Does the graph of $y = -3x^2 + x + 1$ open upwards or downwards?

Solution:

1. The coefficient of $x^2$ is $a = -3$.
2. Since $a < 0$, the parabola opens downwards.

Final Answer: Downwards (∩).

Takeaway: Sign of $a$ decides direction: $a>0$ up, $a<0$ down.

Example 4: Number of zeroes when a parabola touches the axis

A parabola just touches the x-axis at one point. How many zeroes does the quadratic have?

Solution:

1. Touching at one point means the two zeroes are equal (coincident).
2. So there is one distinct zero (a repeated root).

Final Answer: One (repeated) zero.

Takeaway: Touching the axis = equal/coincident zeroes.

Example 5: No real zeroes

If a parabola lies entirely above the x-axis and never meets it, how many real zeroes does the quadratic have?

Solution:

1. Real zeroes correspond to x-intercepts.
2. The graph does not meet the x-axis, so there are no x-intercepts.

Final Answer: Zero real zeroes.

Takeaway: No intercepts ⇒ no real zeroes.

Example 6: Minimum degree from a graph

A graph crosses the x-axis at 3 distinct points. What is the minimum possible degree of the polynomial?

Solution:

1. Number of real zeroes (x-intercepts) = 3.
2. Degree must be at least the number of zeroes.

Final Answer: Minimum degree is 3.

Takeaway: Degree ≥ number of x-intercepts.

Example 7: Reading zeroes from intercepts

A quadratic graph cuts the x-axis at $x = -1$ and $x = 4$. What are its zeroes?

Solution:

1. Zeroes are the x-coordinates of the intercepts.
2. So the zeroes are $-1$ and $4$.

Final Answer: $-1$ and $4$.

Takeaway: Read the zeroes directly off the x-axis crossings.

Example 8: Zeroes of $y = x^2 - 9$ graphically

Where does $y = x^2 - 9$ meet the x-axis?

Solution:

1. Set $y = 0$: $x^2 - 9 = 0$.
2. $x^2 = 9 \Rightarrow x = 3$ or $x = -3$.
3. So the parabola cuts the x-axis at $(3,0)$ and $(-3,0)$.

Final Answer: At $x = 3$ and $x = -3$ (two zeroes).

Takeaway: Solving $p(x)=0$ gives the x-intercepts.

Example 9: Shape of a linear graph

What is the shape of the graph of $y = 5x + 1$, and how many times does it cross the x-axis?

Solution:

1. A degree-1 (linear) polynomial graphs as a straight line.
2. A non-horizontal straight line crosses the x-axis exactly once.

Final Answer: A straight line; it crosses the x-axis once.

Takeaway: Linear ⇒ straight line ⇒ exactly one zero.

Example 10: Maximum zeroes of a cubic

A cubic polynomial's graph is drawn. What is the maximum number of points at which it can meet the x-axis?

Solution:

1. A cubic has degree 3.
2. The graph can cross the x-axis at most 3 times.

Final Answer: At most 3 points.

Takeaway: Maximum x-intercepts = degree = 3 for a cubic.