Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Algebra ↓


Identities and Simplification


Introduction to algebraic identities

Algebraic identities are equations that are true for all values of the variables involved. They are like mathematical truths that we can use to simplify complex algebraic expressions. Understanding these identities helps us solve algebra problems more easily. These identities form the backbone of many algebraic manipulations and simplifications. Let's dive deep into the world of algebraic identities and learn how they work.

General algebraic identities

There are several standard algebraic identities that you will encounter often. Here are some of the most common identities:

1. Identification for classes

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

When you square a binomial, it expands into three terms: the square of the first term, twice the product of the first and second terms, and the square of the second term.

2. Identification of reduced squares

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

This identity is similar to the first one, but it has a subtraction sign. The difference is seen in the middle term, which becomes negative.

3. Difference of squares

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

This identity expresses the difference of two squares as the product of the sum and the difference. It is very useful when factoring expressions.

4. Identifying cubes

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

This identity shows how you can expand the cube of a quantity in four terms.

5. Identifying reduced cubes

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

Similar to the previous identity but involving subtraction, affecting the signs of the resulting terms.

6. Sum of cubes

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

This identity helps in factoring the sum of two cubes.

7. Difference of cubes

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

It is similar to the sum of cubes, but it expresses the difference of cubes.

Simplification using identities

Now that you know some common identities, let's look at how they can be used to simplify algebraic expressions. Simplification involves reducing an expression to its simplest form. This can be done by applying suitable identities.

Example: Simplify (x + 3) 2

                (x + 3) 2 Apply the square identity: (a + b) 2 = a 2 + 2ab + b 2 = x 2 + 2 * x * 3 + 3 2 = x 2 + 6x + 9
                (x + 3) 2 Apply the square identity: (a + b) 2 = a 2 + 2ab + b 2 = x 2 + 2 * x * 3 + 3 2 = x 2 + 6x + 9

Example: Simplify (4y - 1)(4y + 1)

                (4y - 1)(4y + 1) Apply the difference of squares identity: a 2 - b 2 = (a - b)(a + b) Let a = 4y and b = 1 = (4y) 2 - 1 2 = 16y 2 - 1
                (4y - 1)(4y + 1) Apply the difference of squares identity: a 2 - b 2 = (a - b)(a + b) Let a = 4y and b = 1 = (4y) 2 - 1 2 = 16y 2 - 1

Example: Simplify x 3 + 27

                x 3 + 27 is a sum of cubes Apply the sum of cubes identity: a 3 + b 3 = (a + b)(a 2 - ab + b 2) Let a = x and b = 3 = (x + 3)(x 2 - 3x + 9)
                x 3 + 27 is a sum of cubes Apply the sum of cubes identity: a 3 + b 3 = (a + b)(a 2 - ab + b 2) Let a = x and b = 3 = (x + 3)(x 2 - 3x + 9)

Visualizing algebraic identities

Let's take a visual approach to understanding how these identities work. Looking at the geometric representation of these algebraic identities can make them more intuitive.

Visual example: (a + b) 2 = a 2 + 2ab + b 2

one 2 Now Now B2

This figure shows the expansion of (a + b) 2 as a large square divided into four parts: a 2, ab, ab, and b 2. The two ab rectangles represent the middle term: 2ab.

Visual example: a 2 - b 2 = (a + b)(a - b)

one 2 B(AB) B(AB)

In this visualization of the difference of squares identity, the green square a 2 has had b 2 (the pink square) removed, and is replaced by two rectangles representing the product b(ab).

Practical applications of identities

Algebraic identities are not just theoretical constructs; they have practical applications in various fields such as engineering, data analysis, computer science, and others. Here's how understanding algebraic identities can be useful:

  • Problem Solving: Algebraic identities help simplify complex expressions, making them easier to analyze and solve.
  • Programming: When writing code, software developers use algebraic identities to optimize algorithms for faster calculations.
  • Engineering: Engineers use identities to simplify the equations governing physical phenomena, thereby helping to design and operate systems efficiently.
  • Finance: In finance, identities help simplify expressions in financial models, making calculations more straightforward.

Practice problems

Here are some practice problems to solidify your understanding of algebraic identities:

  1. Simplify the expression (2x + 5) 2 using standard identities.
  2. Factor y 2 - 9 using the identity of the difference of squares.
  3. Simplify (m - 6) 3 using the cube identity.
  4. Use identities to expand (3p + 4)(3p - 4).
  5. Factor out x 3 - 8 using the identity of the difference of cubes.

Solve these problems by using identities as a shortcut to make the calculations easier.

Conclusion

Algebraic identities are powerful tools that help us simplify and efficiently manipulate algebraic expressions. They enable us to break down complex problems into manageable parts, leading to easier calculations and solutions. Mastering these identities gives you a solid foundation in algebra and prepares you for more advanced mathematical concepts. Keep practicing, and you'll soon find that these identities will become second nature as you solve more algebraic problems.


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