Polynomials · Lesson 1 of 4

Polynomials

“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 meaning of a polynomial. • Identify the terms, coefficients, variable and constant term of a polynomial. • Classify polynomials according to their degree and number of terms. • Understand the meaning of a zero of a polynomial. • Check whether a given number is a zero of a polynomial.

Introduction to Polynomials

Algebra helps us describe quantities that can change. Instead of using only fixed numbers, we use letters such as x, y, or a to represent values that may vary.

For example, suppose the side of a square is x metres. Its area is x² square metres. If the side is increased by 3 metres, the new side becomes x + 3, so the new area is (x + 3)².

Expressions like x², x + 3, and (x + 3)² are examples of algebraic expressions. Many algebraic expressions are made by combining variables and numbers using addition, subtraction, and multiplication. A very important group of such expressions is called polynomials.

Polynomials appear naturally in problems involving areas, distances, patterns, and changing quantities. Understanding them helps us describe relationships mathematically and prepares us for solving more advanced algebraic problems.

Area of a Square with Side x + 3LaTeX

(x + 3)^{2} = (x + 3) (x + 3) = x^{2} + 6 x + 9

The expression x² + 6x + 9 is an example of a polynomial.

Definition

Polynomial

A polynomial in one variable is an algebraic expression in which the powers of the variable are non-negative whole numbers such as 0, 1, 2, 3 and so on.

General Form of a PolynomialLaTeX

p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{2} x^{2} + a_{1} x + a_{0}

The coefficients a₀, a₁, a₂, ..., aₙ are real numbers, and aₙ must not be zero.

Allowed Powers

The powers of the variable in a polynomial may be 0, 1, 2, 3 and so on. Negative powers, fractional powers and variables in denominators are not allowed.

Expression	Polynomial?	Reason
5x² + 3x − 7	Yes	The powers of x are 2, 1 and 0.
x⁴ − 2x + 9	Yes	All powers of x are non-negative whole numbers.
7	Yes	A constant can be written as 7x⁰.
1/x + 3	No	1/x = x⁻¹, which has a negative power.
√x + 2	No	√x = x¹⁄², which has a fractional power.
2ˣ + 1	No	The variable appears in the exponent.

Important

Coefficients may be positive, negative, fractional, decimal or irrational. The restriction applies to the powers of the variable, not to the coefficients.

Parts of a Polynomial

A polynomial is made up of several parts. To understand these parts, consider the polynomial p(x) = 5x³ − 2x² + 7x − 4.

Example PolynomialLaTeX

p (x) = 5 x^{3} - 2 x^{2} + 7 x - 4

Addition and subtraction signs separate the polynomial into individual terms.

Definition

Variable

A symbol whose value can change is called a variable. In the polynomial 5x³ − 2x² + 7x − 4, the variable is x.

Definition

Terms

The parts of a polynomial separated by addition or subtraction signs are called its terms. The terms are 5x³, −2x², 7x and −4.

Definition

Coefficient

The numerical factor multiplying a variable is called its coefficient. The coefficients of x³, x² and x are 5, −2 and 7 respectively.

Definition

Constant Term

A term that does not contain a variable is called the constant term. In this polynomial, the constant term is −4.

Definition

Degree

The highest power of the variable with a non-zero coefficient is called the degree of the polynomial. The degree of 5x³ − 2x² + 7x − 4 is 3.

Part	Value
Variable	x
Terms	5x³, −2x², 7x, −4
Coefficient of x³	5
Coefficient of x²	−2
Coefficient of x	7
Constant term	−4
Degree	3

Missing Terms

If a term is missing, its coefficient is zero. For example, in x³ + 4x − 5, the coefficient of x² is 0.

Worked Example: Identifying the Parts

Problem
Identify the terms, coefficients, constant term and degree of 4x⁴ − 3x² + 8.

1.The terms are 4x⁴, −3x² and 8.
2.The coefficient of x⁴ is 4.
3.The coefficient of x² is −3.
4.The coefficients of x³ and x are 0 because those terms are missing.
5.The constant term is 8.
6.The highest power of x is 4.
7.Therefore, the degree of the polynomial is 4.

Common Mistake

Do not confuse the coefficient with the degree. In 100x² + x⁵, the largest coefficient is 100, but the degree is 5 because 5 is the highest power of x.

Types of Polynomials

Polynomials can be classified in two common ways: according to their degree and according to the number of non-zero terms they contain.

Classification According to Degree

Type	Degree	General Form	Example
Constant polynomial	0	c	7
Linear polynomial	1	ax + b	2x − 3
Quadratic polynomial	2	ax² + bx + c	x² + 5x + 6
Cubic polynomial	3	ax³ + bx² + cx + d	2x³ − x + 4

Why Must the Leading Coefficient Be Non-Zero?

In the general forms above, the coefficient of the highest-power term must not be zero. For example, in ax² + bx + c, a ≠ 0. Otherwise, the x² term would disappear and the polynomial would not be quadratic.

Classification According to Number of Terms

Type	Number of Terms	Example
Monomial	One non-zero term	5x³
Binomial	Two non-zero terms	x² − 4
Trinomial	Three non-zero terms	x² + 5x + 6

Two Classifications at the Same Time

A polynomial can have one classification based on degree and another based on the number of terms. For example, x² − 4 is both a quadratic polynomial and a binomial.

Worked Example: Classifying a Polynomial

Problem
Classify 3x³ − 5x according to its degree and number of terms.

1.The highest power of x is 3.
2.Therefore, it is a cubic polynomial.
3.It contains two non-zero terms: 3x³ and −5x.
4.Therefore, it is also a binomial.
5.Hence, 3x³ − 5x is a cubic binomial.

Zeroes of a Polynomial

A polynomial gives an output for every value substituted in place of its variable. Some input values make the output exactly zero. These special input values are called the zeroes of the polynomial.

Definition

Value of a Polynomial

If x is replaced by a real number k in p(x), the resulting value is written as p(k).

Definition

Zero of a Polynomial

A real number k is called a zero of the polynomial p(x) if p(k) = 0.

Condition for a ZeroLaTeX

p (k) = 0

The input k is a zero only when the output of the polynomial is exactly zero.

Polynomial as an Input-Output Machine

Think of a polynomial as a machine. You enter a value of x, the machine performs the operations in the polynomial and gives an output p(x). Any input that produces the output 0 is called a zero.

Worked Example: Checking a Zero

Problem
Check whether 2 is a zero of p(x) = x² − 5x + 6.

1.Substitute x = 2 in the polynomial.
2.p(2) = 2² − 5(2) + 6
3.p(2) = 4 − 10 + 6
4.p(2) = 0
5.Since p(2) = 0, the number 2 is a zero of the polynomial.

VerificationLaTeX

p (x) p (2) = x^{2} - 5 x + 6 = 2^{2} - 5 (2) + 6 = 4 - 10 + 6 = 0

Since the output is zero, 2 is a zero of the polynomial.

Worked Example: A Number That Is Not a Zero

Problem
Check whether 1 is a zero of p(x) = x² − 5x + 6.

1.Substitute x = 1 in the polynomial.
2.p(1) = 1² − 5(1) + 6
3.p(1) = 1 − 5 + 6
4.p(1) = 2
5.Since p(1) is not equal to zero, 1 is not a zero of the polynomial.

Worked Example: Finding Two Zeroes by Checking Values

Problem
Show that −1 and 4 are zeroes of p(x) = x² − 3x − 4.

1.First substitute x = −1.
2.p(−1) = (−1)² − 3(−1) − 4 = 1 + 3 − 4 = 0.
3.Therefore, −1 is a zero.
4.Now substitute x = 4.
5.p(4) = 4² − 3(4) − 4 = 16 − 12 − 4 = 0.
6.Therefore, 4 is also a zero.
7.Hence, the zeroes of the polynomial are −1 and 4.

Verification of the ZeroesLaTeX

p (- 1) p (4) = (- 1)^{2} - 3 (- 1) - 4 = 0 = 4^{2} - 3 (4) - 4 = 0

Both inputs produce an output of zero.

Zero of a Linear Polynomial

To find the zero of a linear polynomial, set the polynomial equal to zero and solve the resulting equation.

Worked Example: Zero of a Linear Polynomial

Problem
Find the zero of p(x) = 3x − 12.

1.Set the polynomial equal to zero.
2.3x − 12 = 0
3.3x = 12
4.x = 4
5.Verification: p(4) = 3(4) − 12 = 0.
6.Therefore, the zero of the polynomial is 4.

Zero of a Linear PolynomialLaTeX

a x + b a x x = 0 = - b = - \frac{b}{a}

Every linear polynomial ax + b, where a ≠ 0, has exactly one zero.

Be Careful with Negative Values

Always place a negative substituted value inside brackets. For example, when x = −2, write (−2)². Without brackets, the sign may be handled incorrectly.

Value and Zero Are Different

The value p(3) is simply the output obtained when x = 3. The number 3 is called a zero only when p(3) = 0.

Common Mistakes

• Assuming that every algebraic expression is a polynomial. • Allowing negative or fractional powers of the variable. • Confusing the largest coefficient with the degree. • Forgetting that missing terms have coefficient zero. • Forgetting brackets while substituting negative values. • Calling a number a zero without checking whether it makes the polynomial equal to zero.

Quick check

Which of the following is a polynomial?

Quick check

What is the degree of 8x⁵ − 3x² + 7?

Quick check

How is x² − 4 classified?

Quick check

When is a real number k called a zero of p(x)?

Quick check

What is the zero of the linear polynomial 2x + 6?

Practice Questions

State whether 4x³ − 2x + 7 is a polynomial.
State whether √x + x² is a polynomial.
Identify the terms, coefficients and constant term of 6x³ − 4x² + 9.
Find the degree of 8x⁵ − 3x² + 1.
Classify x² − 9 according to its degree and number of terms.
Classify 4x³ + 2x² − x according to its degree and number of terms.
Check whether 3 is a zero of p(x) = x² − 4x + 3.
Check whether −2 is a zero of p(x) = x² + 5x + 6.
Find the zero of the linear polynomial 5x + 10.
Find the zero of the linear polynomial 7x − 21.

Key Takeaways

• A polynomial contains variables raised only to non-negative whole-number powers. • The main parts of a polynomial are its variable, terms, coefficients, constant term and degree. • Polynomials may be classified according to their degree or their number of terms. • Degree 1, 2 and 3 polynomials are called linear, quadratic and cubic polynomials. • A polynomial with one, two or three non-zero terms is called a monomial, binomial or trinomial. • A real number k is a zero of p(x) when p(k) = 0. • The zero of ax + b is −b/a, where a ≠ 0.

Start of chapter

Next · Lesson 2

Understand the geometrical meaning of a zero

Introduction to Polynomials

Parts of a Polynomial

Types of Polynomials

Classification According to Degree

Classification According to Number of Terms

Zeroes of a Polynomial

Zero of a Linear Polynomial

Quiz

Practice Problems

Key Takeaways