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Lesson 2 of 4

Polynomials · Lesson 2 of 4

Understand the geometrical meaning of a zero

“Mastering polynomials is all about checking the variable vibe rules, finding the highest power in the expression, and revealing how equations secretly control geometric shapes.”

Learning Objectives

• Understand the geometrical meaning of a zero. • Identify zeroes from the graph of a polynomial. • Understand the graphs of linear, quadratic and cubic polynomials. • Determine the number of zeroes by counting the points where a graph meets the x-axis. • Understand the relationship between the degree of a polynomial and its maximum possible number of zeroes.

Geometrical Interpretation of Zeroes

In the previous lesson, we learned that a number k is a zero of a polynomial p(x) when p(k) = 0. We can also understand zeroes visually by drawing the graph of y = p(x) on a Cartesian plane.

Core Principle

The zeroes of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x) intersects or touches the x-axis.

Every point on the x-axis has y-coordinate 0. Therefore, whenever the graph of y = p(x) meets the x-axis, we have p(x) = 0. The corresponding x-coordinate is a zero of the polynomial.

Zero as an x-InterceptLaTeX
At an x-intercept, the y-coordinate is zero.
Crossing and Touching Both Count

A graph may cross the x-axis or only touch it and turn back. In both situations, the point of contact represents a zero because its y-coordinate is 0.

Graph of a Linear Polynomial

The graph of a linear polynomial y = ax + b, where a ≠ 0, is a straight line. Such a line intersects the x-axis at exactly one point. Therefore, a linear polynomial has exactly one zero.

Zero of a Linear PolynomialLaTeX
The x-coordinate of the point where the line meets the x-axis is the zero of the polynomial.
Worked Example: Graph of y = 2x + 3

Problem
Find the zero of p(x) = 2x + 3 and interpret it geometrically.

  1. 1.To find the point where the graph meets the x-axis, put y = 0.
  2. 2.2x + 3 = 0
  3. 3.2x = −3
  4. 4.x = −3/2
  5. 5.Therefore, the graph meets the x-axis at the point (−3/2, 0).
  6. 6.The x-coordinate −3/2 is the zero of the polynomial.
x-Intercept of y = 2x + 3LaTeX
The algebraic answer and the graphical x-intercept are the same.
Linear Polynomial

A non-constant linear polynomial has exactly one zero because its straight-line graph meets the x-axis exactly once.

Graph of y = 2x + 3 x y -5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 y = 2x + 3 (-1.5, 0) At the x-intercept, y = 0: 0 = 2x + 3 → x = -1.5
Graph of y = 2x + 3

Graph of a Quadratic Polynomial

The graph of a quadratic polynomial y = ax² + bx + c, where a ≠ 0, is a symmetrical curve called a parabola.

ConditionDirectionShape
a > 0The parabola opens upwardsU-shaped
a < 0The parabola opens downwardsInverted U-shaped

Depending on its position, a parabola can meet the x-axis at two points, one point or no point. Therefore, a quadratic polynomial can have two, one or no real zeroes.

Number of ZeroesBehaviour of the GraphGeometrical Meaning
Two distinct zeroesThe parabola cuts the x-axis at two different points.Two different x-coordinates
One zeroThe parabola touches the x-axis at exactly one point and turns back.One x-coordinate
No real zeroesThe parabola remains completely above or below the x-axis.No x-intercept

Case 1: Two Distinct Zeroes

When a parabola cuts the x-axis at two different points, the x-coordinates of those points are the two distinct zeroes of the polynomial.

Example: y = x² − 3x − 4

Problem
Identify the zeroes of p(x) = x² − 3x − 4.

  1. 1.Factorise the polynomial.
  2. 2.x² − 3x − 4 = (x − 4)(x + 1)
  3. 3.The polynomial becomes zero when x = 4 or x = −1.
  4. 4.Therefore, the graph cuts the x-axis at (4, 0) and (−1, 0).
  5. 5.Hence, the polynomial has two distinct zeroes: −1 and 4.
Two Distinct ZeroesLaTeX
The two factors produce two different x-intercepts.

Case 2: One Zero

Sometimes a parabola touches the x-axis at only one point and then turns back. In this case, the two algebraic zeroes are equal, so the graph has only one distinct x-intercept.

Example: y = x² − 4x + 4

Problem
How many zeroes does p(x) = x² − 4x + 4 have?

  1. 1.Factorise the polynomial.
  2. 2.x² − 4x + 4 = (x − 2)²
  3. 3.The polynomial becomes zero when x = 2.
  4. 4.Both factors give the same zero.
  5. 5.The parabola touches the x-axis at (2, 0).
  6. 6.Therefore, the polynomial has one distinct zero.
One Repeated ZeroLaTeX
The zero x = 2 occurs twice algebraically but gives one point on the graph.

Case 3: No Real Zeroes

If a parabola remains completely above or completely below the x-axis, it does not have any x-intercept. Therefore, the polynomial has no real zeroes.

Example: y = x² + 1

Problem
How many real zeroes does p(x) = x² + 1 have?

  1. 1.For every real value of x, x² is greater than or equal to 0.
  2. 2.Therefore, x² + 1 is always greater than or equal to 1.
  3. 3.The value of the polynomial can never become zero.
  4. 4.Its graph remains above the x-axis.
  5. 5.Therefore, the polynomial has no real zeroes.
Quadratic Polynomial Summary

A quadratic polynomial can have at most two real zeroes. Its parabola may cut the x-axis twice, touch it once or not meet it at all.

Real Roots of a Quadratic (Parabola) For f(x) = ax² + bx + c, the sign of the discriminant b² − 4ac decides how many times the curve crosses the x-axis 0 Real Roots Discriminant < 0 — parabola never touches the x-axis -4 -3 -2 -1 1 2 3 4 -4 4 8 12 x y 0 vertex no real roots b² − 4ac = 0.8² − 4(0.4)(3) = −4.16 < 0 1 Real Root Discriminant = 0 — vertex sits exactly on the x-axis -4 -3 -2 -1 1 2 3 4 -4 4 8 12 x y 0 vertex x = 0.5 (double) b² − 4ac = (−0.4)² − 4(0.4)(0.1) = 0 2 Real Roots Discriminant > 0 — parabola crosses the x-axis twice -4 -3 -2 -1 1 2 3 4 -4 4 8 12 x y 0 vertex x = −2 x = 3 b² − 4ac = (−0.4)² − 4(0.4)(−2.4) = 4.0
3 types of Roots of A Quadratic Equation

Graph of a Cubic Polynomial

The graph of a cubic polynomial y = ax³ + bx² + cx + d can bend and change direction. It can meet the x-axis at a maximum of three distinct points. Therefore, a cubic polynomial can have at most three zeroes.

Example: y = x³ − 4x

Problem
Find the zeroes of p(x) = x³ − 4x and interpret them geometrically.

  1. 1.Take x as a common factor.
  2. 2.x³ − 4x = x(x² − 4)
  3. 3.Use the identity x² − 4 = (x − 2)(x + 2).
  4. 4.Therefore, p(x) = x(x − 2)(x + 2).
  5. 5.The polynomial becomes zero when x = −2, 0 or 2.
  6. 6.The graph meets the x-axis at (−2, 0), (0, 0) and (2, 0).
  7. 7.Therefore, the polynomial has three distinct zeroes.
Three Zeroes of a Cubic PolynomialLaTeX
Each factor gives one x-intercept.
Example: y = x³

Problem
How many distinct zeroes does p(x) = x³ have?

  1. 1.Set x³ equal to zero.
  2. 2.x³ = 0 gives x = 0.
  3. 3.The graph meets the x-axis only at the origin (0, 0).
  4. 4.Therefore, the polynomial has one distinct real zero.
One Zero of y = x³LaTeX
The curve passes through the origin and has only one distinct x-intercept.
Real Roots of a Cubic Polynomial A cubic ax³+bx²+cx+d always has at least one real root — so the possible cases are 1, 2, or 3 real roots 1 Real Root Monotonic — no turning points -3 -2 -1 1 2 3 -10 -5 5 10 x y 0 x = −1 1 Real Root Turning points, but both above the x-axis -3 -2 -1 1 2 3 -10 -5 5 10 x y 0 x ≈ −2.28 2 Distinct Real Roots One repeated (double) root -3 -2 -1 1 2 3 -10 -5 5 10 x y 0 x = −1 (double) x = 2 3 Distinct Real Roots Three separate crossings -3 -2 -1 1 2 3 -10 -5 5 10 x y 0 x = −2 x = 0 x = 2
Maximum Does Not Mean Exact

The degree gives the maximum possible number of zeroes. It does not mean that every cubic polynomial must have three distinct zeroes.

Degree and Maximum Number of Zeroes

Degree Bound Rule

A polynomial of degree n can have at most n zeroes. Graphically, the graph of y = p(x) can meet the x-axis at no more than n distinct points.

Maximum Number of ZeroesLaTeX
The actual number of real zeroes may be smaller than the degree.
Polynomial TypeDegreeMaximum Number of Zeroes
Linear1At most 1 zero
Quadratic2At most 2 zeroes
Cubic3At most 3 zeroes
Polynomial of degree nnAt most n zeroes
How to Count Zeroes from a Graph

To find the number of zeroes from a graph, count the number of distinct points where the graph intersects or touches the x-axis. Do not count where it meets the y-axis.

Visual Identification

Path 1: Graph Parallel to the x-Axis

Problem
A horizontal graph lies completely above the x-axis. How many zeroes does it have?

  1. 1.The graph does not touch or cross the x-axis.
  2. 2.Therefore, there is no point with y = 0.
  3. 3.Hence, the graph has zero zeroes.
Path 2: Parabola Touching the x-Axis

Problem
An inverted parabola has its highest point at (3, 0). How many zeroes does it have?

  1. 1.The parabola touches the x-axis at exactly one point.
  2. 2.The x-coordinate of that point is 3.
  3. 3.Therefore, the polynomial has one distinct zero: x = 3.
Path 3: Graph Meeting the x-Axis Four Times

Problem
A graph intersects the x-axis at four distinct points. How many zeroes does the polynomial have?

  1. 1.Each distinct x-intercept represents one zero.
  2. 2.The graph has four distinct x-intercepts.
  3. 3.Therefore, the polynomial has four zeroes.
  4. 4.Its degree must be at least 4.
Common Mistakes

• Counting the points where the graph meets the y-axis instead of the x-axis. • Counting only crossings and forgetting that touching the x-axis also represents a zero. • Assuming that a quadratic polynomial must always have two real zeroes. • Assuming that a polynomial of degree n must have exactly n real zeroes. • Counting the same touching point twice when the zero is repeated.

Quiz

Quick check

What do the zeroes of a polynomial represent geometrically?

Quick check

How many zeroes does a non-constant linear polynomial have?

Quick check

How many real zeroes can a quadratic polynomial have?

Quick check

What is the maximum number of zeroes of a cubic polynomial?

Quick check

What are the zeroes of p(x) = x³ − 4x?

Quick check

What does it mean when a parabola touches the x-axis at exactly one point?

Practice Problems

Practice Questions
  1. A graph crosses the x-axis at x = −3 and x = 5. Write its zeroes.
  2. A parabola touches the x-axis only at x = 4. How many distinct zeroes does it have?
  3. A quadratic graph remains completely above the x-axis. How many real zeroes does it have?
  4. The graph of a cubic polynomial meets the x-axis at x = −2, 1 and 3. State its zeroes.
  5. Can a quadratic polynomial have three distinct zeroes? Give a reason.
  6. Can a polynomial of degree 5 have six distinct zeroes? Give a reason.
  7. Find the x-intercept of the graph y = 3x − 6.
  8. Find the zeroes of y = x² − 9 and state the corresponding x-intercepts.
  9. How many zeroes does y = x² + 4 have?
  10. A graph touches the x-axis at one point and crosses it at two other points. How many distinct zeroes does it have?

Key Takeaways

Key Takeaways

• The zeroes of p(x) are the x-coordinates where the graph of y = p(x) meets the x-axis. • A graph may cross the x-axis or touch it. Both situations represent zeroes. • A non-constant linear polynomial has exactly one zero. • A quadratic polynomial may have two, one or no real zeroes. • A cubic polynomial can have at most three zeroes. • A polynomial of degree n can have at most n zeroes. • To count zeroes from a graph, count its distinct x-intercepts.

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Polynomials

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Relationship between Zeroes and Coefficients of a Polynomial