Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Algebra → Polynomials ↓


Algebraic Identities


Algebraic identities are equations that are valid for all values of the variables involved. They are important tools in algebra used to simplify polynomial expressions and solve equations more quickly. When you learn about algebraic identities, you will discover patterns that hold true under a variety of circumstances, which can often save time and effort in calculations.

Understanding algebraic identities

Before diving into specific algebraic identities, it's important to understand the fundamentals of polynomials. A polynomial is an expression consisting of variables (often called indeterminates) and coefficients that are combined using addition, subtraction, multiplication, and non-negative integer exponents. Here's a simple polynomial expression:

ax² + bx + c

In this expression, a, b, and c are coefficients, and x is the variable. The highest exponent in a polynomial determines its degree.

General algebraic identities

Let's look at some common algebraic identities. These include squares of sums, squares of differences, and products of sums and differences, etc.

1. The square of a sum

The identity of the square of a quantity is expressed as follows:

(a + b)² = a² + 2ab + b²

This means that if you take the sum of two terms and square it, it will be equal to the square of the first term, twice the product of the two terms, and the square of the second term.

Let's understand this identity visually:

A² Now Now B²

Visually, (a + b)² is represented as a large square made up of four regions: a small square of size a², two rectangles of size ab, and a smaller square b².

2. The square of the difference

The identity of the square of the difference is:

(a - b)² = a² - 2ab + b²

Unlike addition, this identity holds for the case where the squaring involves subtraction, and results in twice the product of the two terms being subtracted.

Here's a visual example for (a - b)²:

A² -Now -Now B²

3. Product of sum and difference

This identity is expressed as follows:

(a + b)(a – b) = a² – b²

This simplifies the difference of squares. This is a very useful identity because it allows the product of a sum and a difference to be reduced directly to the difference of squares.

Example

Consider the expression (5 + 3)(5 - 3) According to the identity:

(5 + 3)(5 - 3) = 5² - 3² = 25 - 9 = 16

4. Cube of the sum

When it comes to cubing a quantity, the expression is more complicated. The identity is:

(a + b)³ = a³ + 3a²b + 3ab² + b³

This identity shows that when you cube a sum, the result is a polynomial with four terms.

5. Cube of the difference

Similarly, the identity for the cube of the difference is:

(a - b)³ = a³ - 3a²b + 3ab² - b³

As with squaring, subtracting terms changes some of the signs in the polynomial.

Applications of algebraic identities

Algebraic identities are powerful tools that simplify the process of expanding and factoring polynomials. Recognizing these identities helps in a variety of mathematical calculations and problem-solving scenarios. Here are some applications:

Simplification of expressions

Identities make simplifying expressions easier. Instead of expanding polynomials term by term, you can quickly apply identities to get the result.

Example

Simplify (x + 4)²

(x + 4)² = x² + 2 * 4 * x + 4² = x² + 8x + 16

Factorization of polynomials

Recognizing identities helps break down polynomials into simpler components, making them easier to solve or manipulate.

Example

Factor x² - 16.

x² – 16 = (x)² – (4)² = (x + 4)(x – 4)

Solving equations

Algebraic identities can make equations easier to solve by providing substitutions that simplify the equation.

Example

Find the solution of x in x² + 10x + 25 = 0.

x² + 10x + 25 = (x + 5)² = 0
x + 5 = 0
x = -5

Why learn algebraic identities?

Learning algebraic identities is important as they provide a deeper understanding of polynomial structures. Recognizing these patterns is helpful not only in academic settings but also in fields involving computations such as computer science, engineering, physics, etc.

Practicing algebraic identities

To become proficient at algebraic identities, constant practice is essential. Try creating your own expressions and verifying them using identities. Practice identifying them and applying them in different scenarios. Here are some exercises to help you get started:

  • Simplify: (x + 2)²
  • Simplify: (3a - 2b)²
  • Factor: a² - 9b²
  • Simplify: (2x + 3y)(2x - 3y)
  • Simplify: (2a + 3b)³

Conclusion

Algebraic identities are the foundational pillars in the study of algebra. They simplify complex expressions and equations, making mathematical problem-solving more efficient. Understanding and applying these identities is crucial to mastering algebra and advancing in mathematics. By practicing these identities and incorporating them into everyday math problems, students can enhance their analytical skills and mathematical understanding.


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Factorization of Polynomials
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Algebraic Identities

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