Polynomials

Why This Chapter Matters

Algebra becomes powerful when we stop treating expressions as strings of symbols and start seeing their structure. A polynomial is one of the simplest structured algebraic objects: terms are built from non-negative whole-number powers of a variable and then combined by addition or subtraction.

Why are polynomials important? They model patterns, areas, motion and equations. More importantly, they teach a central mathematical habit: evaluate, test, factor and reason from structure.

The chapter moves naturally from recognising polynomials to asking a deeper question: when does a polynomial become zero? That question leads to the remainder theorem, factor theorem and factorisation. The whole chapter is therefore a story about how a complicated expression reveals hidden factors when we test the right values.

Bridge from Earlier Classes

Before starting, be comfortable with:

• Algebraic expressions
• Like and unlike terms
• Laws of exponents
• Class 8 identities
• Basic factorisation

These are the tools this chapter reuses and deepens.

1. What Counts as a Polynomial?

A polynomial in x contains terms such as 5x³, -2x and 7, where the exponents of x are non-negative integers.

Expressions such as 3/x, √x or x^-2 are not polynomials in x because they involve negative or fractional powers. The restriction on exponents is what gives polynomials their predictable algebraic behaviour.

The coefficient is the numerical factor of a term. A missing power can be thought of as having coefficient zero. For example, x³-5x+2 can be written as x³+0x²-5x+2.

2. Degree and Types

The degree of a non-zero polynomial is the highest exponent with a non-zero coefficient. A non-zero constant polynomial has degree 0, a linear polynomial degree 1, a quadratic degree 2 and a cubic degree 3.

The zero polynomial needs special care. Since every coefficient is zero, there is no highest power with a non-zero coefficient, so its degree is not assigned in the usual school definition.

3. Value and Zeros

To evaluate p(x) at x=a, substitute a everywhere x occurs. If p(a)=0, then a is called a zero of the polynomial.

For a linear polynomial ax+b, the zero is -b/a when a≠0. A zero is not a missing term or a zero coefficient. It is an input value that makes the entire polynomial equal to zero.

This distinction is crucial because zeros later become x-coordinates where graphs meet the x-axis and also create factors through the factor theorem.

4. Remainder Theorem

Suppose p(x) is divided by x-a. The remainder theorem says the remainder is p(a).

Why is this useful? Polynomial division can be long, while substitution is quick. If the divisor is x-3, evaluate p(3). If the divisor is x+2, rewrite it as x-(-2), so evaluate p(-2).

The theorem creates a bridge between division and evaluation. It is also the immediate foundation for the factor theorem.

5. Factor Theorem

The factor theorem is the zero-remainder case of the remainder theorem. If p(a)=0, dividing by x-a leaves remainder zero, so x-a is a factor. Conversely, if x-a is a factor, p(a)=0.

This suggests a systematic strategy for factorising higher-degree polynomials: test likely integer values, find a zero, remove the corresponding linear factor, then factor the remaining lower-degree expression.

6. Factorisation by Identities

Identities are algebraic relationships that hold for all permitted values. They are useful in both directions: expansion and factorisation.

Before applying an identity, look for structure. Are the first and last terms perfect squares? Is the middle term twice the product? Is the expression a difference of squares? Is there a common factor that should be taken out first?

A reliable routine is: common factor → identity pattern → factor theorem if needed.

7. Identities as Calculation Tools

Identities are not only examination formulas. They simplify numerical calculations as well. For example, 103² can be evaluated as (100+3)² instead of using long multiplication.

The deeper lesson is that symbolic algebra can make arithmetic more efficient by exposing structure.

Visualise the Core Idea

A polynomial becomes easier when you separate its structural parts.

Formula / Theorem Sheet

Worked Examples — Follow the Reasoning

How to Think in the Exam

Common Mistakes

Quick Revision

Focused Practice

Final Chapter Summary