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 ↓


Algebraic Identities


Algebraic identities are equations that are true for all values of the variables within them. They serve as helpful shortcuts when working with polynomials. Polynomials are mathematical expressions that contain variables and coefficients, which are combined using addition, subtraction, multiplication, and non-negative integer exponents.

These identities help us simplify polynomial expressions, solve equations, and perform calculations faster without having to manually multiply or expand the expressions each time.

Basic algebraic identities

Let us look at some of the fundamental algebraic identities that are most common in Class 9 Maths:

  • Square of the sum:

The square of the sum identity states:

(a + b) 2 = a 2 + 2ab + b 2

Visual example:

A² B² Now Now (a+b)²

In this identity, a and b are variables or constants, and their squares a² and b² are represented by small squares, while 2ab is represented by two rectangles.

Example calculation:

(5 + 3) 2 = 5 2 + 2 * 5 * 3 + 3 2 = 25 + 30 + 9 = 64
  • Square of the difference:

Similar to the square of a sum, we also have the square of a difference:

(a - b) 2 = a 2 - 2ab + b 2

Example calculation:

(7 - 2) 2 = 7 2 - 2 * 7 * 2 + 2 2 = 49 - 28 + 4 = 25
  • Product of sum and difference:

This identity is useful for multiplying polynomials of the form sum and difference:

(a + b)(a - b) = a 2 - b 2

This implies that the product of the sum and the difference of the same two terms is the difference of squares.

Example calculation:

(8 + 3)(8 - 3) = 8 2 - 3 2 = 64 - 9 = 55

More advanced algebraic identities

Although the following identities in basic polynomial operations are less common, they are still very useful:

  • Cube of the sum:

The cube of a quantity is expressed as:

(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3

Example calculation:

(2 + 4) 3 = 2 3 + 3 * 2 2 * 4 + 3 * 2 * 4 2 + 4 3 = 8 + 48 + 96 + 64 = 216
  • Cube of the difference:

The cube of the difference is drawn as follows:

(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3

Example calculation:

(5 - 3) 3 = 5 3 - 3 * 5 2 * 3 + 3 * 5 * 3 2 - 3 3 = 125 - 75 + 45 - 27 = 68
  • Sum or difference of cubes:

The identities for the sum and difference of cubes are given as follows:

a 3 + b 3 = (a + b)(a 2 - ab + b 2 )

a 3 - b 3 = (a - b)(a 2 + ab + b 2 )

Example of calculating the sum of cubes:

4 3 + 2 3 = (4 + 2)(4 2 - 4*2 + 2 2 ) = 6(16 - 8 + 4) = 6 * 12 = 72

Example calculation for difference of cubes:

5 3 - 3 3 = (5 - 3)(5 2 + 5*3 + 3 2 ) = 2(25 + 15 + 9) = 2 * 49 = 98

Applications of algebraic identities

Algebraic identities simplify the process of expanding polynomials and solving equations. When dealing with large numbers or complex expressions, these identities reduce the time and effort of calculations.

Consider the polynomial x² + 6x + 9 Recognizing it as a perfect square, we can immediately write it as (x + 3)² instead of multiplying it manually.

For example, using algebraic identities you can solve:

(3x + 5)² = 9x² + 30x + 25

This reduces to:

3x + 5 = 0

Simplifying this further, you get:

x = -5/3

The importance of practicing algebraic identities

Practicing algebraic identities helps students develop an intuitive understanding of mathematical structures, thereby enhancing problem-solving skills. These skills become foundational as students progress in mathematics and form the basis for many advanced mathematical concepts.

Regularly practicing these identities through exercises such as expanding and factoring polynomial expressions will make these rules become natural and improve mathematical fluency.

Conclusion

Understanding and using algebraic identities aids in the simplification and efficient calculation of polynomial expressions. By mastering basic and advanced identities, students can solve mathematical problems quickly and accurately, laying the groundwork for more complex topics in mathematics.

Through continued practice and application of these identities, students will enhance their algebraic thinking, gaining valuable skills for future learning and problem-solving contexts.


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