Roots of Polynomials

Introduction: Why do roots matter? 🌱

Hello students, in this lesson you will learn about roots of polynomials, a key idea in the study of functions. A polynomial is a function made from terms like $x^2$, $x^3$, or $x^n$, combined using addition, subtraction, and multiplication by constants. Roots are the values of $x$ that make the polynomial equal to $0$. In other words, if $f(x)$ is a polynomial and $f(a)=0$, then $a$ is a root of the polynomial.

Why is this important? Because roots tell us where a graph crosses or touches the $x$-axis, which helps us solve equations, sketch graphs, and understand real situations such as profit models, motion, and population change 📈. In IB Mathematics: Analysis and Approaches HL, you need to connect algebraic methods with graphical meaning, so roots are a perfect example of how functions work as both formulas and visual objects.

Learning objectives

By the end of this lesson, students, you should be able to:

explain the main ideas and vocabulary connected to roots of polynomials,
find roots using algebraic and graphical reasoning,
connect roots to factors, intercepts, multiplicity, and transformations,
use roots in equations and inequalities involving functions,
summarize why roots are a central idea in the topic of Functions.

What is a polynomial root?

A polynomial is a function of the form $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_n\neq 0$ and the exponents are whole numbers. The degree of the polynomial is the highest power of $x$, which is $n$.

A root or zero of the polynomial is any number $r$ such that $f(r)=0$. On a graph, this is the point where the function meets the $x$-axis, so the $y$-value is $0$.

For example, if $f(x)=x^2-5x+6$, then:

$$f(2)=2^2-5(2)+6=0$$

so $x=2$ is a root. Also,

$$f(3)=3^2-5(3)+6=0$$

so $x=3$ is another root.

This means the graph of $f(x)=x^2-5x+6$ crosses the $x$-axis at $x=2$ and x=3`.

Factors, roots, and the Factor Theorem

Roots are closely linked to factors. If $f(r)=0$, then $(x-r)$ is a factor of $f(x)$. This is the Factor Theorem.

For the example above,

$$x^2-5x+6=(x-2)(x-3)$$

so the factors directly show the roots $x=2$ and $x=3$.

This idea is extremely useful because many polynomial questions can be solved by changing between different forms:

expanded form: $f(x)=x^2-5x+6$,
factorised form: $f(x)=(x-2)(x-3)$.

Factorised form makes roots visible immediately. Expanded form is often useful for calculation and substitution. Being able to move between the two is an important IB skill.

A quick example with a cubic:

$$g(x)=x^3-4x^2-x+4$$

Try testing small integer values such as $x=1$:

$$g(1)=1-4-1+4=0$$

So $(x-1)$ is a factor. After factorising, you may find more roots. This process combines numerical checking, algebra, and structure.

Multiplicity: crossing or touching the axis

Not every root behaves the same way on a graph. This is where multiplicity matters.

If a factor appears once, such as $(x-a)$, then $a$ is a root of multiplicity $1$. The graph usually crosses the $x$-axis at that root.

If a factor appears twice, such as $(x-a)^2$, then $a$ is a root of multiplicity $2$. The graph usually touches the $x$-axis and turns around there.

If a factor appears three times, such as $(x-a)^3$, the graph crosses the axis but flattens at the intercept.

For example,

$$h(x)=(x-1)^2(x+2)$$

has roots $x=1$ and $x=-2$.

$x=1$ has multiplicity $2$, so the graph touches the $x$-axis there.
$x=-2$ has multiplicity $1$, so the graph crosses the $x$-axis there.

This is very useful in graph sketching because it helps you predict the shape of the polynomial without calculating many points.

How many roots can a polynomial have?

A polynomial of degree $n$ can have at most $n$ roots. This is a fundamental result in algebra.

For example:

a quadratic polynomial has degree $2$, so it can have at most $2$ roots,
a cubic polynomial has degree $3$, so it can have at most $3$ roots,
a quartic polynomial has degree $4$, so it can have at most $4$ roots.

These roots may be real or complex. In IB Functions, you often focus on real roots because they are the ones you can see on a graph as $x$-intercepts.

A polynomial may have fewer than its degree in real roots. For example,

$$p(x)=x^2+1$$

has no real roots because $x^2+1=0$ has no real solution. Its roots are complex, $x=\pm i$, but those do not appear as $x$-intercepts on the real graph.

This is important because it shows that “having roots” depends on the number system you are using. In real graphing, the key question is whether $f(x)=0$ has real solutions.

Finding roots using algebraic methods

There are several ways to find polynomial roots.

1. Factorising

If a polynomial factorises nicely, set each factor equal to $0$.

Example:

$$x^2-9=0$$

Factorise:

$$(x-3)(x+3)=0$$

So the roots are $x=3$ and $x=-3$.

2. Using the zero product rule

$$(x-a)(x-b)=0$$

then either $x-a=0$ or $x-b=0$.

This rule is the reason factorisation works so well.

3. Substitution and testing rational roots

For higher-degree polynomials, you may test possible rational roots based on the constant term and leading coefficient. For example, if

$$f(x)=2x^3-3x^2-8x+12$$

you might test values like $x=1,-1,2,-2,3,-3,\frac{1}{2}$, and so on.

If $f(2)=0$, then $(x-2)$ is a factor. After dividing by $(x-2)$, you reduce the problem to a quadratic or another easier polynomial.

This strategy is common in IB because it shows logical reasoning, not just memorizing formulas.

4. Graphing technology

A graphing calculator or software can show approximate roots. If a root is difficult to find exactly, the graph may help you estimate it.

For example, if

$$q(x)=x^3-x-1$$

there is no simple factorisation. A graph may show one real root near $x=1.3$. Technology is useful here, but you still need to understand what the root means and how to verify it.

Roots in equations and inequalities involving functions

Roots are not only about graphs; they are also about solving equations and inequalities.

If you solve

$$f(x)=0$$

you are finding the roots.

If you solve

$$f(x)>0$$

$$f(x)<0$$

you are using the roots to find intervals where the function is positive or negative.

For example, let

$$f(x)=(x-1)(x+2)$$

The roots are $x=1$ and $x=-2$.

Now consider the inequality

$$(x-1)(x+2)>0$$

The product is positive when both factors are positive or both are negative. This happens on the intervals

$$x<-2 \quad \text{or} \quad x>1.$$

So the solution is

$$(-\infty,-2)\cup(1,\infty).$$

This shows why roots are so important in inequality problems: they divide the number line into regions where the function changes sign.

Connection to the broader topic of Functions

Roots of polynomials fit into Functions because they connect algebra, graphs, and function language.

A function is a rule that assigns each input $x$ a unique output $f(x)$. The roots tell you which inputs produce output $0$. This means roots are one of the most direct ways to study a function’s behavior.

Roots also help with:

graph interpretation: roots are $x$-intercepts,
transformations: shifting a polynomial changes its roots,
composite and inverse ideas: solving $f(x)=0$ is like finding inputs mapped to a special output,
modeling: roots can represent break-even points in business or moments when a quantity becomes zero.

For example, if a profit function is $P(x)$, then solving $P(x)=0$ gives break-even points. In a motion problem, roots might show when an object hits the ground.

Because IB Mathematics: Analysis and Approaches HL emphasizes relationships between algebraic expressions and graphs, roots are a core part of the topic of Functions.

Conclusion

Roots of polynomials are the values of $x$ that make a polynomial equal to $0$. They can be found by factorising, using the zero product rule, testing possible values, or graphing. Roots are linked to factors, intercepts, and inequalities, and their multiplicity tells you how the graph behaves at the $x$-axis.

students, understanding roots helps you solve equations, sketch polynomial graphs, and interpret real-world situations. In the IB course, this idea is not isolated: it connects directly to function language, graph behavior, and algebraic reasoning. Mastering roots gives you a strong foundation for more advanced function topics later on 🔎.

Study Notes

A root or zero of a polynomial is a value $r$ such that $f(r)=0$.
Roots are the same as $x$-intercepts on the graph of a function when the root is real.
If $f(r)=0$, then $(x-r)$ is a factor of $f(x)$ by the Factor Theorem.
A polynomial of degree $n$ has at most $n$ roots.
Roots can have multiplicity:
multiplicity $1$: graph usually crosses the $x$-axis,
multiplicity $2$: graph usually touches and turns,
multiplicity $3$: graph crosses with flattening.
To solve $f(x)=0$, try factorising, using the zero product rule, testing rational roots, or graphing.
For inequalities like $f(x)>0$ or $f(x)<0$, first find the roots, then test intervals.
Roots connect polynomial algebra to graphical behavior, which is central to the Functions topic.
Real-world meaning: roots can represent break-even points, zero height, or any point where a quantity becomes $0$.
In IB Mathematics: Analysis and Approaches HL, roots are essential for reasoning, sketching, and interpreting polynomial functions.