Polynomials: Know Remainder Theorem

Polynomials are a core part of algebra and show up again and again in exams, so understanding them clearly can boost both speed and confidence in problem‑solving.​ Let's explore.

What is a Polynomial?

A polynomial is an algebraic expression made up of variables and constants where the powers of the variables are non‑negative integers, and the terms are combined using addition, subtraction, and multiplication. For example, 

• 2x2+3x−5

• 2x

• 2

• +3x−5 is a polynomial,

but expressions like 

• x

• x

•  or 

• x−1

• x

• −1 are not polynomials because their exponents are not whole numbers.​

Types and Degree of Polynomials (Monomial, Binomial, Quadratic etc.)

Polynomials are often classified in two ways: by the number of terms and by their degree, which helps in selecting the right method for solving and factorization. A monomial has one term, a binomial has two, and a trinomial has three, while the degree tells you if it is linear, quadratic, cubic, or biquadratic based on the highest power of the variable.​
Zeros of polynomials and why they matter
The zeros of a polynomial are the values of the variable for which the polynomial becomes zero, and they are closely linked to the degree of the polynomial. A polynomial of degree 

• n

• n can have at most 

• n

• n zeros, and finding these zeros helps in factorization, graphing, and solving equations quickly in exam questions.​

Remainder Theorem for Quick Remainders

The remainder theorem gives a fast shortcut for finding the remainder when a polynomial 

• p(x)

• p(x) is divided by a linear expression of the form 

• x−a

• x−a without doing full long division. According to this theorem, the remainder is simply 

• p(a)

• p(a), which means you just substitute 

• a

• a into the polynomial and evaluate, saving time during exams.​

Factor Theorem and Factorization of Polynomials

The factor theorem is a powerful tool that connects zeros and factors: if 

• p(a)=0

• p(a)=0, then 

• x−a

• x−a is a factor of the polynomial, and if 

• x−a

• x−a is a factor, then 

• a

• a is a zero. Using this idea with methods like middle‑term splitting, special identities, and hit and trial, you can break a polynomial into simpler factors that are easier to work with and verify by multiplying back.​

Important Algebraic Identities You Must Know

Algebraic identities such as 

• (a+b)2

• (a+b)

• (a−b)2

• (a−b)^2

• (a+b)(a−b)

• (a+b)(a−b), and formulas for cubes and sum or difference of cubes are invaluable shortcuts when expanding or factorizing polynomials. Learning identities like 

• (a+b+c)2

• (a+b+c)^2

•  and special cube identities can turn long multiplications into quick steps and are frequently used in exam‑oriented polynomial questions.​

Exam Tips for Mastering Polynomials Quickly

For exam preparation, focus on recognizing whether an expression is a polynomial, identifying its degree, and deciding quickly which technique to use—remainder theorem, factor theorem, or algebraic identities. Practice factorization of quadratic and higher‑degree polynomials, verify answers by multiplying the factors, and skip lengthy proofs so you can spend more time on application and problem‑solving.