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


Factoring is a concept in algebra that involves breaking down an expression into a product of simpler expressions, called factors. When you factor an expression, you are essentially rewriting it as a multiplication problem. are. Factoring is a powerful tool in algebra because it helps simplify expressions, solve equations, and make complex calculations more manageable.

What is factorization?

Factoring is the process of expressing an algebraic expression as a product of its factors. These factors can be numbers, variables, or a combination of both. For example, the expression x^2 + 5x + 6 (x + 2)(x + 3). Here, (x + 2) and (x + 3) are the factors of the original expression.

Why is factorization important?

Factoring is important for several reasons:

  • It simplifies complex algebraic expressions.
  • It helps in solving quadratic and polynomial equations.
  • It is used in dividing polynomials.
  • It helps in finding the zeros or roots of polynomial functions.

Fundamentals of factorization

To factor an algebraic expression, you need to follow certain principles and techniques. Let's explore some basic concepts of factorization.

1. Common factors

A common factor is a number or variable that is a factor of every term in an expression. The first step in factoring is to identify any common factors in the terms of the expression.

6x^2 + 9x Common factor: 3x Factorized form: 3x(2x + 3)

2. Factoring by grouping

When there is no common factor for all terms, but the expression can be grouped into pairs with common factors, we can use factorization by grouping.

x^2 + 5x + 6 Grouping: (x^2 + 3x) + (2x + 6) Common factors: x(x + 3) + 2(x + 3) Factorized form: (x + 2)(x + 3)

3. Difference of squares

Difference of squares is a specific pattern where an expression can be factored when it is of the form a^2 - b^2, resulting in (a + b)(a - b).

x^2 - 9 Recognize as: x^2 - 3^2 Factorized form: (x + 3)(x - 3)
a^2 - b^2 A+B A - B

Factoring quadratic expressions

Standard form

A quadratic expression usually has the form ax^2 + bx + c. To factor it, we need to find two numbers whose product is ac and whose sum is b.

Example of factoring quadratic

Factor x^2 + 5x + 6.

Find two numbers whose product is 6 (coefficient of x^2 * constant term) and sum is 5 (coefficient of x). The numbers are 2 and 3. Rewriting: x^2 + 2x + 3x + 6 Group: (x^2 + 2x) + (3x + 6) Factor common: x(x + 2) + 3(x + 2) Factorized form: (x + 2)(x + 3)

Special factoring techniques

Perfect square trinomial

Some quadratics are perfect squares. This means that they can be expressed as the square of a binomial.

x^2 + 4x + 4 Recognize as: (x + 2)^2

Sum/difference of cubes

Cubes also have specific factorization formulas. The sum of cubes a^3 + b^3 factors (a + b)(a^2 - ab + b^2), and the difference of cubes a^3 - b^3 factors (a - b)(a^2 + ab + b^2).

x^3 + 27 Recognize as: x^3 + 3^3 Factorized form: (x + 3)(x^2 - 3x + 9)
a^3 + b^3 (a + b) (a^2 - ab + b^2)

Practice problems

  1. Factor 3x^2 + 12x.
  2. Factor y^2 - 16.
  3. Factor 2x^2 + 7x + 3.
  4. Factor 64a^3 - 27b^3.
  5. Factor x^2 - 6x + 9.

Suggested techniques:

  • First, look at the common factors.
  • Check whether the expression is a perfect square or a difference of squares.
  • For quadratics, find numbers that multiply by ac and add to b.
  • Use specific formulas for cubes.

Conclusion

Factoring is an important concept in algebra that allows us to simplify expressions, solve equations, and understand the properties of polynomials. By mastering the techniques of factoring, you can gain powerful tools to work effectively with algebraic expressions. Get the tools. Remember, practice is key to becoming proficient at factoring, so use the examples and techniques discussed here as a basis for further exploration and problem-solving.


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Factorization

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