Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Polynomials ↓


Definition and Types of Polynomials


In mathematics, polynomials play a vital role and are essential in understanding various algebraic expressions and equations. Here, you will learn what polynomials are, their different types, and various examples associated with them. This lesson will guide you through a simple explanation of polynomials suitable for grade 9 students.

What is a polynomial?

A polynomial is an algebraic expression made up of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. Here's what a polynomial typically looks like:

a n x n + a n-1 x n-1 + ... + a 1 x + a 0

In this expression:

  • a n , a n-1 , ..., a 1 , a 0 are the coefficients.
  • x is variable or undefined.
  • n is the degree of the polynomial, which is the highest power of the variable in the polynomial.

Let's look at a simple example:

3x 2 + 2x + 1

Here, 3x 2 + 2x + 1 is a polynomial of degree 2, where:

  • 3 is the coefficient of x 2.
  • 2 is the coefficient of x.
  • 1 is a constant term.

Types of polynomials

Polynomials can be classified into different types depending on the number of terms and the degree of the polynomial.

Based on the number of terms

Polynomials are classified into the following types depending on the number of terms they contain:

  • Monomial: A polynomial with only one term. For example: 
    5x 3
  • Binomial: A polynomial with two terms. For example: 
    7x + 4
  • Trinomial: A polynomial with three terms. For example: 
    x 2 + 3x + 2

Depending on the degree of the polynomial

The degree of a polynomial is the highest power of the variable present in the polynomial. The types based on the degree are as follows:

  • Zero polynomial: A polynomial whose all coefficients are equal to zero. It has no degree. For example: 
    0
  • Constant polynomial: A polynomial of degree 0 that has only one constant term and no variables. For example: 
    7
  • Linear polynomial: A polynomial of degree 1. Its graph is a straight line. For example: 
    3x + 4
  • Quadratic polynomial: A polynomial of degree 2. When graphed it forms a parabola. For example: 
    x 2 + 2x + 1
  • Cubic polynomial: A polynomial of degree 3. It may have one or two turning points when graphed. For example: 
    2x 3 - 4x 2 + x + 5
  • Quartic polynomial: A polynomial of degree 4. For example: 
    x 4 - x 3 + 2x - 1
  • Quinary polynomial: A polynomial of degree 5. For example: 
    x 5 - 3x 4 + x 3 - x + 1

Visualization of polynomials

Let's visualize some polynomials to understand them better. We will start with simple linear polynomials and move on to more complex polynomials.

y = 2x + 1

The red line above represents a linear polynomial, 2x + 1 Notice how it forms a straight line.

Now, let's look at a quadratic polynomial, which forms a curve called a parabola.

y = x² - x

The blue curve represents a quadratic polynomial, x² - x See how it forms a 'U' shape.

More examples of polynomials

In addition to the basics and visual examples, let's work through some problems and practice recognizing different types of polynomials.

Example 1: Determine the type of the polynomial:

5x 4 + 3x 2 - x + 1

This polynomial has four terms. It is a quartic polynomial because its degree is 4, which is the largest exponent.

Example 2: Identify and classify the polynomial:

7y - 9

This polynomial has two terms. It is a linear binomial because its degree is 1 (the highest power of y), and it has two terms.

Example 3: Write a polynomial and classify it:

Let's write a polynomial with three terms, the degree of which is 3.

2x 3 + x 2 - 4

This is a trinomial because it has three terms, and the highest power is 3.

Conclusion

In this lesson, we explored the definition and classification of polynomials based on the number of terms and their degree. Polynomials form the building blocks of algebra and are essential in everything from solving equations to graphing functions.

Understanding the different types of polynomials helps students develop a strong foundation in algebra and prepares them to tackle more complex mathematical concepts in the future. Keep practicing with different examples to increase your understanding of how these algebraic expressions work!


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Definition and Types of Polynomials

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