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 → Factorization ↓


Special Products


Factorization is an important concept in algebra that involves expressing a mathematical expression as a product of its factors. In this context, we talk about "special products," which are special forms or patterns of algebraic expressions. These special patterns often appear in algebra, and recognizing them can greatly simplify the process of factoring.

In 8th grade math, learning about special products helps students simplify expressions and solve equations more efficiently. Let's dig deeper into these special products, breaking down several concepts to make it easier to understand. With visual and textual examples.

Square of binomial

A binomial is an algebraic expression consisting of two terms, such as a + b or a - b. The square of a binomial is a common special product, and it follows a specific pattern. The general formula for squaring a binomial is:

(a + b)² = a² + 2ab + b²

Similar to:

(a - b)² = a² - 2ab + b²

These formulas mean that when you square a binomial, you get three terms: the square of the first term, twice the plus/minus of the product of the two terms, and the square of the second term.

A² Now Now B²

The diagram above shows the formula (a + b)² = a² + 2ab + b². Here, a² is the area of the yellow square, ab is the area of the red and orange rectangles, and b² is the area of the green square.

Examples of squaring a binomial

Let's consider some examples of how this rule applies:

(x + 3)² = x² + 2 * x * 3 + 3² = x² + 6x + 9

Note that according to our formula: a² = x², 2ab = 6x, and b² = 9.

(2y - 5)² = (2y)² - 2 * 2y * 5 + 5² = 4y² - 20y + 25

In this case, we have a = 2y, and b = 5 Therefore, a² = 4y², 2ab = -20y, and b² = 25.

Product of sum and difference

Another important special product is the result of multiplying the sum of two terms by their difference. The general formula is:

(a + b)(a - b) = a² - b²

This formula tells us that when you multiply a sum and its conjugate difference, the result is a difference of squares.

A² -B²

The diagram shows a² - b². Let's apply the formula to some concrete numbers.

Examples of the product of the sum and difference

(x + 4)(x - 4) = x² - 4² = x² - 16

In this example, a = x and b = 4 Therefore, this is simplified to x² - 16.

(3y + 2)(3y - 2) = (3y)² - 2² = 9y² - 4

Here, a = 3y and b = 2, which simplifies 9y² - 4.

Cube of a binomial

The concept of cubing a binomial is an extension of classification. For any binomial, cubing follows a recognizable pattern. The cube of a binomial is:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Alternatively, if we reverse the addition:

(a - b)³ = a³ - 3a²b + 3ab² - b³

This pattern emerges from applying the principles of binomial expansion to the third power.

Examples of cubic binomials

(x + 2)³ = x³ + 3x²*2 + 3x*2² + 2³ = x³ + 6x² + 12x + 8

In this calculation, each term is derived from the binomial expansion pattern.

(y - 5)³ = y³ - 3y²*5 + 3y*5² - 5³ = y³ - 15y² + 75y - 125

Here too, the cube of a binomial formula shows its usefulness, and efficiently breaks down a complex expression into manageable parts.

Illustrating the cube of a binomial

″ 3a²b 3AB² B³

The visual representation above helps to understand the components of the expanded cube of a binomial. Understanding how each part fits into the whole cube can help to understand the whole concept more effectively.

Conclusion

Special products represent important shortcuts in algebra, derived from regular patterns found when multiplying binomials. Understanding these fundamental concepts is essential for simplifying expressions and solving polynomial equations. Such special products, including perfect square trinomials, difference of squares, and cubes of binomials, are widely applied, not only within algebra, but also in more advanced mathematical contexts.

Mastering these products can lead to a deeper understanding of mathematical structures and relationships, boosting the ability to see complex patterns in simpler forms. Use these patterns to your advantage in solving everyday algebra problems. Use this as a basis for more advanced studies.


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