Types of Polynomials - Overview, Structure, Properties & Uses

Types of Polynomials Based on Degree of Polynomials

Polynomials are commonly classified based on their degree, which is the highest power of the variable in the expression. Understanding types of polynomials based on degree is essential for solving algebraic problems and questions in competitive exams.

Classification of polynomials based on degree

Type of Polynomial	Description	Example
Constant (Zero) Polynomial	Degree is $0$ and contains only a constant term	$P(x) = 2$, $P(x) = -5$
Linear Polynomial	Degree is $1$	$P(x) = 8x$, $P(x) = x + 2$
Quadratic Polynomial	Degree is $2$	$x^2 + 2x + 1$, $2x^2$
Cubic Polynomial	Degree is $3$	$x^3 + 2x + 1$, $6x^3$
Biquadratic Polynomial	Degree is $4$	$2x^4 + x^2 + 1$

These types of polynomial functions are widely used in algebra, equations, and graph-based problems.

Types of Polynomial Based on Number of Terms

Polynomials can also be classified based on the number of terms present in the expression. This classification is important for identifying the structure of algebraic expressions.

Monomial

A monomial is a polynomial with only one term.

It can contain one or more variables
No addition or subtraction within the term

Examples:

$x^2$ (one variable)
$x^2 y$ (two variables)
$x^2 y z^3$ (three variables)

Monomials are the simplest form of polynomial expressions.

Binomial

A binomial is a polynomial with exactly two terms.

Terms are separated by + or − sign
Can include one or more variables

Examples:

$x^2 + 3x$
$x^2 + 3y$
$x^2 + 3yz^3$

Binomials are commonly used in algebraic identities and factorization.

Trinomial

A trinomial is a polynomial with three terms.

Terms are connected using addition or subtraction
Frequently used in quadratic expressions

Examples:

$x^2 + 3x + 2$
$x^2 + 3y + 1$
$x^2 + 3y + z^3$

Trinomials are important for solving quadratic equations and factorization problems.

Special Types of Polynomials

Apart from standard classifications, there are some special types of polynomials that are important in advanced algebra and competitive exams.

Homogeneous Polynomial

A polynomial in which all terms have the same degree.

Each term has equal total power
Useful in algebra and higher mathematics

Example:

$x^2 + y^2 + z^2$

Irreducible Polynomial

A polynomial that cannot be factorized further over a given field.

Cannot be broken into simpler polynomial factors
Important in algebraic theory

Example:

$x^2 + 3$

Monic Polynomial

A polynomial in which the leading coefficient (coefficient of highest degree term) is $1$.

Simplifies many algebraic calculations
Common in polynomial equations

Example:

$x^2 + 3x + 1$

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Properties of Polynomials

Polynomials have several important properties and theorems that help in solving algebraic equations, factorisation problems, and competitive exam questions efficiently. Understanding these properties of polynomials in algebra is essential for mastering concepts like division, roots, and polynomial behaviour.

Division Algorithm for Polynomials

The division algorithm is a fundamental concept used in polynomial division.

If $f(x)$ is the dividend and $g(x) \neq 0$ is the divisor, then there exist unique polynomials $q(x)$ (quotient) and $r(x)$ (remainder) such that:

$f(x) = g(x) \cdot q(x) + r(x)$

The degree of $r(x)$ is less than the degree of $g(x)$
If $r(x) = 0$, then $f(x)$ is exactly divisible by $g(x)$

This property is widely used in solving polynomial division questions.

Bezout’s Theorem in Polynomials

Bezout’s theorem connects polynomials with their greatest common divisor.

For two polynomials $f(x)$ and $g(x)$, there exist polynomials $a(x)$ and $b(x)$ such that:

$f(x) \cdot a(x) + g(x) \cdot b(x) = \gcd(f(x), g(x))$

Any expression of this form is a multiple of the GCD
Useful in solving higher-level algebra problems

Remainder Theorem

The remainder theorem is one of the most important results in polynomial algebra.

If a polynomial $f(x)$ is divided by $(x - a)$, then the remainder is:

$f(a)$

This can also be written as:

$f(x) = (x - a)g(x) + f(a)$

Helps in quickly finding remainders without long division
Commonly used in competitive exam questions

Factor Theorem

The factor theorem is closely related to the remainder theorem.

A polynomial $f(x)$ has $(x - a)$ as a factor if and only if:

$f(a) = 0$

If $f(a) = 0$, then $(x - a)$ divides $f(x)$ exactly
Useful for factorization and finding roots

Intermediate Value Theorem

This theorem explains the behavior of polynomials over an interval.

If $f(x)$ is continuous on $[a, b]$ and $N$ lies between $f(a)$ and $f(b)$, then there exists a value $c \in (a, b)$ such that:

$f(c) = N$

Helps in locating roots of polynomial equations
Important in understanding graph behavior

Algebraic Operations on Polynomials

Polynomials remain closed under basic algebraic operations.

Let $f(x)$ and $g(x)$ be polynomials:

Degree of $f(x) + g(x)$ is less than or equal to the maximum degree
Degree of $f(x) - g(x)$ is less than or equal to the maximum degree
Degree of $f(x) \cdot g(x)$ is the sum of their degrees

These rules are essential for simplifying polynomial expressions.

Divisibility and Zeros of Polynomials

If a polynomial $f(x)$ is divisible by another polynomial $g(x)$, then:

Every zero of $g(x)$ is also a zero of $f(x)$
If $f(a) = 0$, then $a$ is called a zero (root) of the polynomial

Mathematically:

$f(x) = g(x) \cdot r(x)$

This property helps in understanding polynomial roots and factors.

Divisibility by Co-Prime Polynomials

If a polynomial $f(x)$ is divisible by two co-prime polynomials $g(x)$ and $h(x)$, then:

It is divisible by their product

$f(x) = g(x) \cdot h(x)$

Co-prime polynomials have no common factor except $1$

This concept is useful in advanced factorization problems.

Rule of Distinct Roots

A polynomial of degree $n$ can have at most $n$ distinct roots.

If

$f(x) = a_0 + a_1x + a_2x^2 + \dots + a_n x^n$

then it has at most $n$ distinct solutions.

Helps in estimating the number of solutions
Important for solving polynomial equations

Descartes’ Rule of Signs

This rule helps in determining the number of real roots.

The number of positive real roots is equal to the number of sign changes or less than it by an even number
The number of negative real roots is determined by sign changes in $f(-x)$

Example:
$f(x) = 6x^5 + 4x^4 - 2x^3 + x^2 - x - 2$

Sign changes:

$+4x^4 \to -2x^3$
$-2x^3 \to +x^2$
$+x^2 \to -x$

Total sign changes = $3$

So, the number of real roots can be $3$ or $1$

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Advanced Properties and Theorems of Polynomials

Polynomials follow several advanced properties and theorems that are essential for solving higher-level algebra questions and competitive exam problems. These concepts help in understanding roots, factorization, and the behavior of polynomial functions.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is one of the most important results in polynomial theory.

Every non-constant polynomial with complex coefficients has at least one complex root
This means all polynomials can be expressed in terms of their roots

Mathematically:
$f(x) = a_n (x - r_1)(x - r_2)(x - r_3)\dots(x - r_n)$

$a_n$ is the leading coefficient
$r_1, r_2, \dots, r_n$ are roots (real or complex)

This theorem is widely used in solving polynomial equations and factorization.

Rule of Conjugate Roots

This rule applies to polynomials with real coefficients.

If $a + bi$ is a root, then $a - bi$ is also a root
Complex roots always occur in conjugate pairs

Example:
If $2 + 3i$ is a root, then $2 - 3i$ is also a root

This property helps in identifying all roots of a polynomial.

Polynomial Function and Polynomial Equation

This section covers the concept of polynomial functions and equations, helping you understand how polynomials are formed and how they are used to solve algebraic problems.

Polynomial Function

A polynomial function is a function defined by a polynomial expression.

General form:
$f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0$

Examples:

$f(x) = 2x^2 + 5x + 1$ (quadratic)
$f(x) = 8x^3 + 3x + 1$ (cubic)
$f(x) = 2x + 3$ (linear)

Polynomial Equation

A polynomial equation is formed when a polynomial is set equal to zero.

General form:
$f(x) = 0$

Examples:

$2x^2 + 5x + 1 = 0$
$8x^3 + 3x + 1 = 0$
$2x + 3 = 0$

Solving polynomial equations means finding the values of $x$ that satisfy the equation.

Methods to Solve Polynomial Equations

This section introduces step-by-step methods to solve linear, quadratic, and cubic polynomial equations using standard formulas and techniques.

Solving Linear Polynomial

A linear polynomial has degree $1$.

General form:
$f(x) = ax + b$

To solve:
$ax + b = 0 \Rightarrow x = -\frac{b}{a}$

Example:
$5x - 20 = 0 \Rightarrow x = 4$

Solving Quadratic Polynomial

A quadratic polynomial has degree $2$.

General form:
$ax^2 + bx + c = 0$

Methods to solve:

Mid-term splitting
Using sum and product of roots

Formulas:

$\alpha + \beta = -\frac{b}{a}$
$\alpha \beta = \frac{c}{a}$

Example:
$x^2 + 5x + 6 = (x + 2)(x + 3)$

Solving Cubic Polynomial

A cubic polynomial has degree $3$.

General form:
$ax^3 + bx^2 + cx + d = 0$

Key formulas:

$\alpha + \beta + \gamma = -\frac{b}{a}$
$\alpha \beta \gamma = -\frac{d}{a}$
$\alpha \beta + \beta \gamma + \alpha \gamma = \frac{c}{a}$

Example:
For $6x^3 + 2x^2 + x + 1$

Sum of roots = $-\frac{1}{3}$

Operations on Polynomials

This section explains basic operations like addition, subtraction, multiplication, and division of polynomials with examples for better understanding. Polynomials follow standard algebraic operations.

Addition of Polynomials

Add like terms (same powers)

Example:
$6xy(2x-4z) + 3yz(2x-3z) + 4xz(3y-2y^2)$
$= 12x^2y - 6xyz - 9yz^2 - 8xy^2z$

Subtraction of Polynomials

Subtract corresponding terms

Example:
$(6x^3 + 2x^2 + 5) - (4x^3 + x^2 - 5) = 2x^3 + x^2 + 10$

Multiplication of Polynomials

Multiply each term and combine

Example:
$(x + 2)(y + 5) = xy + 2y + 5x + 10$

Division of Polynomials

Use factorization, long division, or synthetic division

Example:
$\frac{x^2 - x - 6}{x + 2} = x - 3$

Understanding these advanced polynomial properties, theorems, and operations helps in solving algebra problems efficiently and improves performance in exams like boards, JEE, and other competitive tests.

Tips and Tricks for Polynomial Problems

This section covers important shortcuts and key formulas that help in solving polynomial questions quickly and accurately in exams. These tips are useful for understanding standard forms, performing operations, and applying root-based formulas efficiently.

Standard forms of polynomials

Linear polynomial: $ax + b$
Quadratic polynomial: $ax^2 + bx + c$
Cubic polynomial: $ax^3 + bx^2 + cx + d$
Biquadratic polynomial: $ax^4 + bx^3 + cx^2 + dx + e$

Knowing these standard forms helps in quickly identifying the type and degree of a polynomial.

Rules for polynomial operations

Add or subtract only like terms with the same degree
Combine coefficients carefully to avoid errors
In multiplication, multiply each term of one polynomial with every term of the other
Always simplify the result by combining like terms

These basic rules are essential for solving algebraic expressions correctly.

Formulas for quadratic polynomials

For the quadratic equation $ax^2 + bx + c = 0$, if the roots are $\alpha$ and $\beta$:

Sum of roots: $\alpha + \beta = -\frac{b}{a}$
Product of roots: $\alpha \beta = \frac{c}{a}$

These formulas are widely used in solving and forming quadratic equations.

Formulas for cubic polynomials

For the cubic equation $ax^3 + bx^2 + cx + d = 0$, if the roots are $\alpha$, $\beta$, and $\gamma$:

Sum of roots: $\alpha + \beta + \gamma = -\frac{b}{a}$
Product of roots: $\alpha \beta \gamma = -\frac{d}{a}$
Sum of product of roots taken two at a time: $\alpha \beta + \beta \gamma + \alpha \gamma = \frac{c}{a}$

These formulas help in solving higher-degree polynomial equations efficiently.

Important Formulas for Polynomials (Quick Revision Table)

Here is a structured table of the most important polynomial formulas and identities that are frequently used in algebra and competitive exams:

Concept	Formula	Use Case
General form of polynomial	$a_n x^n + a_{n-1} x^{n-1} + \dots + a_0$	Represents any polynomial
Linear polynomial root	$x = -\frac{b}{a}$	Solving $ax + b = 0$
Quadratic standard form	$ax^2 + bx + c$	Identifying quadratic equations
Sum of roots (quadratic)	$\alpha + \beta = -\frac{b}{a}$	Finding relation between roots
Product of roots (quadratic)	$\alpha \beta = \frac{c}{a}$	Used in factorization
Forming quadratic equation	$x^2 - Sx + P$	When sum $S$ and product $P$ are given
Cubic standard form	$ax^3 + bx^2 + cx + d$	Identifying cubic equations
Sum of roots (cubic)	$\alpha + \beta + \gamma = -\frac{b}{a}$	Root relations
Product of roots (cubic)	$\alpha \beta \gamma = -\frac{d}{a}$	Used in solving cubic
Sum of pairwise products	$\alpha \beta + \beta \gamma + \alpha \gamma = \frac{c}{a}$	Cubic root relations
Remainder theorem	Remainder = $f(a)$	When divided by $(x - a)$
Factor theorem	$f(a) = 0$	To check if $(x - a)$ is a factor
Division algorithm	$f(x) = g(x)q(x) + r(x)$	Polynomial division
Identity $(a + b)^2$	$a^2 + 2ab + b^2$	Expansion
Identity $(a - b)^2$	$a^2 - 2ab + b^2$	Expansion
Identity $(a + b)(a - b)$	$a^2 - b^2$	Factorization
Cube identity $(a + b)^3$	$a^3 + 3a^2b + 3ab^2 + b^3$	Higher expansions
Cube identity $(a - b)^3$	$a^3 - 3a^2b + 3ab^2 - b^3$	Higher expansions

Key Takeaways

These formulas are essential for solving polynomial equations quickly
Root-based formulas help in factorization and equation formation
Identities are widely used in simplification and expansion
Regular revision improves speed and accuracy in exams

This table serves as a quick revision guide for all important polynomial formulas.

Best Books for Quantitative Aptitude Preparation

Here is a clean and exam-focused list of the most recommended books for Quantitative Aptitude across SSC, Banking, CAT, and other competitive exams:

Book Name	Author	Key Features	Best For
Quantitative Aptitude for Competitive Examinations	R.S. Aggarwal	Covers all basic to advanced topics with a large number of practice questions	Beginners and SSC aspirants
Fast Track Objective Arithmetic	Rajesh Verma	Focuses on short tricks and faster problem-solving methods	SSC and Banking exams
Quantitative Aptitude for CAT	Arun Sharma	Concept-based learning with difficulty levels for MBA exams	CAT and MBA aspirants
Quantum CAT	Sarvesh K. Verma	Advanced level questions with detailed solutions	CAT and high-level exams
Magical Book on Quicker Maths	M. Tyra	Shortcut techniques and speed improvement methods	Speed enhancement
NCERT Mathematics (Class 11 & 12)	NCERT	Strong foundation building with basic concepts	Beginners and school-level clarity
Objective Arithmetic	S. Chand (Arihant/Other editions)	Topic-wise coverage with practice questions	SSC, Banking, Railways
Data Interpretation & Logical Reasoning	Arun Sharma	Helps with DI along with quant preparation	MBA exams

Practice Questions

Q1. If $x^{2}-3x+1=0$, then the value of $\left(x + \frac{1}{x}\right)$ is:

1
0
3
2

Hint: First, move the $3x$ term to the right and then divide both sides by $x$.

Solution:

Given:
$x^{2}-3x+1=0$

$⇒ x^{2}+1=3x$

$⇒ \frac{x^{2} +1}{x}= 3$

$\therefore x + \frac{1}{x} = 3$

Hence, the correct answer is 3.

Q2. What is the value of $64x^{3} + 36x^{2}y + 24xy^{2} + 2y^{3}$, when $x = 3$ and $y = -4$?

304
1456
1584
432

Hint: Make the equation in the form of $x^{2}(64x + 36y) + y^{2}(24x + 2y)$.

Solution:

Given: $x = 3, y = -4$

$= x^{2}(64x + 36y) + y^{2}(24x + 2y)$

$= 3^{2}[(64 × 3) + (36 × -4)] + (-4)^{2}[(24 × 3) + (2 × -4)]$

$= 9 × [192 -144] + 16 × [72 -8]$

$= (9 × 48) + (16 × 64)$

$= 432 + 1024$

$= 1456$

Hence, the correct answer is $1456$.

Q3. If $x, y,$ and $z$ are the three factors of $a^3-7a-6$, then the value of $x+y+z$ will be:

$3a$
$3$
$6$
$a$

Hint: Split $-7a$ into $(-a-6a)$ to get the factors.

Solution:

Given:
$a^3-7a-6$

$= a^3-a-6a-6$

$= a(a^2-1)-6(a+1)$

$= a(a-1)(a+1)-6(a+1)$

$= (a+1)(a^2-a-6)$

$= (a+1)(a-3)(a+2)$

So, the factors are $(a+1), (a-3), (a+2)$

$\therefore x+y+z= (a+1)+(a-3)+(a+2)=3a$

Hence, the correct answer is $3a$.

Q4. If $(a+b)=5$, then the value of $(a-3)^7+(b-2)^7$ is:

$2^7$
$3^7$
1
0

Hint: Express $a$ in terms of $b$ and substitute.

Solution:

Given: $(a+b)=5$

$⇒ a=5-b$

$= (5-b-3)^7+(b-2)^7$

$= (2-b)^7+(b-2)^7$

$= -(b-2)^7+(b-2)^7$

$= 0$

Hence, the correct answer is $0$.

Q5. If $x^{4}+2x^{3}+ax^{2}+bx+9$ is a perfect square, where $a$ and $b$ are positive real numbers, then the value of $a$ and $b$ are:

$a=5, b=6$
$a=6, b=7$
$a=7, b=7$
$a=7, b=6$

Hint: Assume the expression is $[(x-\alpha)(x-\beta)]^2$.

Solution:

Given:
$x^{4}+2x^{3}+ax^{2}+bx+9$

Let the expression be $[(x-\alpha)(x-\beta)]^2$

So, the roots are $\alpha, \alpha, \beta, \beta$

Sum of roots = $-\frac{\text{coefficient of } x^3}{\text{coefficient of } x^4}$

$⇒ \alpha+\alpha+\beta+\beta = -2$

$⇒ 2(\alpha+\beta) = -2$

$⇒ \alpha+\beta = -1$

Product of roots = $\frac{\text{constant term}}{\text{coefficient of } x^4}$

$⇒ \alpha \cdot \alpha \cdot \beta \cdot \beta = 9$

$⇒ (\alpha\beta)^2 = 9$

$⇒ \alpha\beta = \pm 3$

Now, sum of roots taken two at a time = coefficient of $x^2$

$a = \alpha^2 + \beta^2 + 4\alpha\beta$

$⇒ a = (\alpha+\beta)^2 + 2\alpha\beta$

$⇒ a = (-1)^2 + 2(\pm 3)$

$⇒ a = 1 \pm 6$

Since $a$ is positive,

$⇒ a = 7$

Now, sum of roots taken three at a time = $-\frac{b}{1}$

$⇒ -b = 2\alpha^2\beta + 2\alpha\beta^2$

$⇒ -b = 2\alpha\beta(\alpha+\beta)$

$⇒ -b = 2(3)(-1)$

$⇒ -b = -6$

$⇒ b = 6$

Hence, the correct answer is $a=7, b=6$.

Q6. If $a^3+3 a^2+3 a=63$, then the value of $a^2+2 a$ is:

22
19
15
8

Hint: Use $(a+1)^3=a^3+3a^2+3a+1$.

Solution:

Given:
$a^3+3a^2+3a=63$

Add 1 on both sides:

$a^3+3a^2+3a+1 = 63+1$

$⇒ (a+1)^3 = 64$

Take cube root on both sides:

$a+1 = 4$

$⇒ a = 4 - 1$

$⇒ a = 3$

Now find required value:

$a^2+2a = 3^2 + 2(3)$

$⇒ a^2+2a = 9 + 6$

$⇒ a^2+2a = 15$

Hence, the correct answer is 15.

Q7. If $(x-2)$ is a factor of $x^2+3Qx - 2Q$, then the value of $Q$ is:

2
-2
1
-1

Hint: If $(x-a)$ is a factor, then $P(a)=0$.

Solution:

Given polynomial:

$P(x)=x^2+3Qx-2Q$

Since $(x-2)$ is a factor,

$⇒ P(2)=0$

Substitute $x=2$:

$⇒ 2^2 + 3Q(2) - 2Q = 0$

$⇒ 4 + 6Q - 2Q = 0$

$⇒ 4 + 4Q = 0$

$⇒ 4Q = -4$

$⇒ Q = -1$

Hence, the correct answer is $-1$.

Q8. If $x^{3}+2x^{2}-5x+k$ is divisible by $x+1$, then the value of $k$ is:

-6
-1
0
6

Hint: Use $f(-1)=0$.

Solution:

Given:
$f(x)=x^3+2x^2-5x+k$

Since divisible by $x+1$,

$⇒ f(-1)=0$

Substitute $x=-1$:

$⇒ (-1)^3 + 2(-1)^2 - 5(-1) + k = 0$

$⇒ -1 + 2(1) + 5 + k = 0$

$⇒ -1 + 2 + 5 + k = 0$

$⇒ 6 + k = 0$

$⇒ k = -6$

Hence, the correct answer is $-6$.

Q9. Which of the following equations has 7 as a root?

$3x^2-6x+2=0$
$x^2-9x+14=0$
$x^2-7x+10=0$
$x^2+3x-12=0$

Hint: Substitute $x=7$ or factorize.

Solution:

Check option 1:

$3x^2-6x+2=0$

Substitute $x=7$:

$⇒ 3(49) - 6(7) + 2 = 147 - 42 + 2 = 107 \neq 0$

Not a root.

Check option 2:

$x^2-9x+14=0$

Factorize:

$⇒ x^2 - 7x - 2x + 14 = 0$

$⇒ x(x-7) -2(x-7) = 0$

$⇒ (x-7)(x-2)=0$

$⇒ x=7$ or $x=2$

Thus, 7 is a root.

Hence, the correct answer is $x^2-9x+14=0$.

Q10. One of the factors of the expression $4\sqrt{3}x^{2}+5x-2\sqrt{3}$ is:

$4x+\sqrt{3}$
$4x+3$
$4x-3$
$4x-\sqrt{3}$

Hint: Factorize using middle-term splitting.

Solution:

Given:
$4\sqrt{3}x^{2}+5x-2\sqrt{3}$

Split the middle term:

$= 4\sqrt{3}x^{2} + 8x - 3x - 2\sqrt{3}$

Group terms:

$= (4\sqrt{3}x^{2} + 8x) - (3x + 2\sqrt{3})$

Take common factors:

$= 4x(\sqrt{3}x + 2) - \sqrt{3}(\sqrt{3}x + 2)$

Factor common binomial:

$= (4x - \sqrt{3})(\sqrt{3}x + 2)$

Hence, one factor is $4x - \sqrt{3}$.

Hence, the correct answer is $4x-\sqrt{3}$.

Q11. The term that should be added to $(4x^2+8x)$ so that the resulting expression is a perfect square, is:

2
4
$2x$
1

Hint: Use identity $(a + b)^2 = a^2 + 2ab + b^2$.

Solution:

Given:
$4x^2 + 8x$

Write in identity form:

$4x^2 = (2x)^2$

$8x = 2 \cdot (2x) \cdot 2$

So, expression becomes:

$(2x)^2 + 2(2x)(2)$

To make it a perfect square, add $2^2 = 4$

$⇒ 4x^2 + 8x + 4$

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

$= (2x + 2)^2$

The required term is $4$.

Hence, the correct answer is $4$.

Classification of Numbers	Concept of Relative Speed and Average Speed
Rational Numbers	Time, Speed, Distance
BODMAS Rule in Simplification	Pipe and Cistern and concept of negative work
Factor	Time and Work
Perfect squares and perfect cubes	Application of ratios in partnership
HCF and LCM	Proportions and Variations
Divisibility Rules	Ratio and Proportion
Unit Digit	Loans and Installment
Last Two Digit of a number	Compound Interest
Remainder theorem	Simple Interest
Number of factors and number of trailing zeroes	Profit and Loss
Arithmetic Progression	Application of Percentage
Geometric Progression	Probability
Harmonic Progression	Permutation and Combination
Relation Between AM, GM and HM	Mode
Percentage	Sphere and Hemisphere
Quadratic Equations	Sequence and Series Miscellaneous
Lines and Angles	Triangles and Congruence
Square Root and Cube Root of Surds	Theorem of triangles
Exponent and Surds	Data Interpretation
Linear Equations in Two Variables	Problems related to Data Interpretation
Linear Equations in One Variables	Mean
Maxima and Minima in polynomials	Median
Algebraic Identities	Similar Triangles
Types of Polynomials	Quadrilaterls
Mixture and Alligation	Cyclic Quadrilateral and Geometric Centres
Average	Parallelogram and Mid Point Theorem
Application of TSD in Elevator	Rhombus, Square, Rectangles and Trapezium
Linear Races and Circular Races	Prism
Application of TSD in Trains and Stream	Polygon

Types of Polynomials: Definition, Questions, Formula, Examples

What is a Polynomial in Mathematics?

Definition of polynomial with examples

General form of a polynomial

Key terms: variables, coefficients, constants, degree

Importance of polynomials in algebra and exams

Terms of a Polynomial

Degree of a Polynomial

Types of Polynomials Based on Degree of Polynomials

Classification of polynomials based on degree

Types of Polynomial Based on Number of Terms

Monomial

Binomial

Trinomial

Related eBooks and Sample Papers

Special Types of Polynomials

Homogeneous Polynomial

Irreducible Polynomial

Monic Polynomial

Properties of Polynomials

Division Algorithm for Polynomials

Bezout’s Theorem in Polynomials

Remainder Theorem

Factor Theorem

Intermediate Value Theorem

Algebraic Operations on Polynomials

Divisibility and Zeros of Polynomials

Divisibility by Co-Prime Polynomials

Rule of Distinct Roots

Descartes’ Rule of Signs

Advanced Properties and Theorems of Polynomials

Fundamental Theorem of Algebra

Rule of Conjugate Roots

Polynomial Function and Polynomial Equation

Polynomial Function

Polynomial Equation

Methods to Solve Polynomial Equations

Solving Linear Polynomial

Solving Quadratic Polynomial

Solving Cubic Polynomial

Operations on Polynomials

Addition of Polynomials

Subtraction of Polynomials

Multiplication of Polynomials

Division of Polynomials

Tips and Tricks for Polynomial Problems

Standard forms of polynomials

Rules for polynomial operations

Formulas for quadratic polynomials

Formulas for cubic polynomials

Important Formulas for Polynomials (Quick Revision Table)

Key Takeaways

Best Books for Quantitative Aptitude Preparation

Practice Questions

Related Quantitative Aptitude Topics for Competitive Exams

News and Notifications

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