Polynomials

Polynomials Notes

A polynomial in one variable is an algebraic expression in which the powers of the variable are whole numbers. This chapter covers the degree of a polynomial, zeroes of a polynomial, the Remainder Theorem, the Factor Theorem, factorisation of quadratic and cubic polynomials, and the standard algebraic identities.

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12 practice questions
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Chapter roadmap

Use this map for a first read, then return to the checks and formulas during revision.

1.Polynomials — Terms, Degree, and Types
2.Zeroes of a Polynomial
3.Remainder Theorem and Factor Theorem
4.Algebraic Identities

5 key formulas
2 concept checks

Polynomials — Terms, Degree, and Types

A polynomial in x is an expression of the form aₙxⁿ + … + a₁x + a₀ where the coefficients are real numbers and n is a non-negative integer. Expressions with negative or fractional powers (like x⁻¹ or x^(1/2)) are not polynomials.

The degree is the highest power of the variable with a non-zero coefficient. A polynomial of degree 1 is linear, degree 2 is quadratic, degree 3 is cubic. A non-zero constant has degree 0; the zero polynomial has no defined degree.

The value of a polynomial p(x) at x = a is p(a), obtained by substituting a for x.

Zeroes of a Polynomial

A real number a is a zero (or root) of p(x) if p(a) = 0. Finding zeroes means solving p(x) = 0.

A linear polynomial ax + b has exactly one zero, x = −b/a. A polynomial of degree n has at most n real zeroes. A non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.

Example: for p(x) = x² − 5x + 6, p(2) = 4 − 10 + 6 = 0 and p(3) = 9 − 15 + 6 = 0, so 2 and 3 are the zeroes.

?Check your understanding 1

For which value of

k

x = 2

a zero of

p(x) = 2x^2 + kx + 6

Remainder Theorem and Factor Theorem

Remainder Theorem: if a polynomial p(x) is divided by the linear polynomial (x − a), the remainder is p(a). So you can find the remainder without doing the division.

Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0. More generally, (x − a) is a factor exactly when a is a zero of p(x).

Example: is (x − 1) a factor of p(x) = x³ − 3x² + 3x − 1? p(1) = 1 − 3 + 3 − 1 = 0, so yes. In fact p(x) = (x − 1)³.

p(x) = (x-a)\,q(x) + p(a)

Division of p(x) by (x − a): the remainder is the constant p(a).

?Check your understanding 2

The remainder when

x^3 + 3x^2 + 3x + 1

is divided by

(x + 1)

is:

Algebraic Identities

Identities let you expand and factorise quickly without multiplying out every time. The most used ones in Class 9 are the squares of a binomial, the difference of two squares, the product (x + a)(x + b), and the cubes.

For three variables, (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx is common, and the identity x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx) is used in special factorisations (note that if x + y + z = 0 then x³ + y³ + z³ = 3xyz).

To factorise a quadratic ax² + bx + c, split the middle term: find two numbers whose product is ac and whose sum is b.

(x+y)^3 = x^3 + y^3 + 3xy(x+y)

(x-y)^3 = x^3 - y^3 - 3xy(x-y)

x^3 + y^3 = (x+y)(x^2 - xy + y^2)

x^3 - y^3 = (x-y)(x^2 + xy + y^2)

Polynomials: revision sheet

Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.

Recall checklist

Polynomials — Terms, Degree, and Types
Zeroes of a Polynomial
Remainder Theorem and Factor Theorem
Algebraic Identities

For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.

Formula recap (5)

Remainder Theorem and Factor Theorem

p(x) = (x-a)\,q(x) + p(a)

Division of p(x) by (x − a): the remainder is the constant p(a).

Algebraic Identities

(x+y)^3 = x^3 + y^3 + 3xy(x+y)

Algebraic Identities

(x-y)^3 = x^3 - y^3 - 3xy(x-y)

Algebraic Identities

x^3 + y^3 = (x+y)(x^2 - xy + y^2)

Algebraic Identities

x^3 - y^3 = (x-y)(x^2 + xy + y^2)

Explain your method

Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.

Which of these are polynomials in one variable?

3x^2 - 2x + 5

\sqrt{x} + 1

x^{-2} + x

5

Check worked solution 1

Polynomials:

3x^2 - 2x + 5

and

5

(a constant polynomial of degree 0). Not polynomials:

\sqrt{x}+1

(power 1/2) and

x^{-2}+x

(negative power).

Write the degree of

7 - x^3 + 4x^5 - 2x^2

Check worked solution 2

The highest power is 5, so the degree is 5.

Turn mistakes into revision

Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.

Practise this chapter

Practice and Revision

Test your understanding with quick chapter-level practice.

Open Practice

Chapter Q&A

Is $\sqrt{2}x + 3$ a polynomial?

Yes. The coefficient can be any real number, including an irrational number like √2. What matters is that the power of the variable (here 1) is a whole number.

What is the difference between a zero of a polynomial and the value of a polynomial?

The value of p(x) at x = a is simply p(a). A zero is a special value of x for which p(a) = 0.

How does the Factor Theorem help in factorising a cubic polynomial?

Try small integer values (factors of the constant term). If p(a) = 0, then (x − a) is a factor. Divide the cubic by (x − a) to get a quadratic, which can then be factorised by splitting the middle term.

How many zeroes can a polynomial of degree n have?

At most n real zeroes. A linear polynomial has exactly one; a quadratic has at most two; a cubic has at most three.

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