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 ↓


Factorization by Grouping


Factoring by grouping is a method used to simplify complex expressions by combining terms into groups. By doing this, we make it easier to find common factors and factor the expression completely. This technique is especially useful when dealing with polynomials that have four or more terms.

Basic concept of factorization

Before we get into factoring by grouping, let's understand some basics. Factoring in algebra involves breaking expressions into simpler parts called "factors." For example, the expression:

2x + 6

can be factored as follows:

2(x + 3)

Here, 2 and (x + 3) are factors of the expression 2x + 6.

Steps in factorization by grouping

Factoring by grouping involves several clear steps:

  1. Grouping terms: Break the expression into groups that can be factored separately.
  2. Factor out each group: Factor out the common factors from each group.
  3. Common factor extraction: Look for common factors between groups and extract them as factors.
  4. Simplify: Write the expression in factored form.

Visual understanding of factorization by grouping

Let us explain the steps using a simple example:

Example: x^2 + 5x + 2x + 10 Step 1: Group the words (x^2 + 5x) + (2x + 10) Step 2: Factor each group x(x + 5) + 2(x + 5) Step 3: Take out the common factor (x + 5)(x + 2)

We break down each step, and show how to group and factor the terms until we get a fully factored expression.

Step-by-step example

Now, let's look at another example in more detail:

Example 2: Factor a polynomial

Suppose we have the polynomial:

ax + ay + bx + by

We can factor this expression by following these steps:

Step 1: Group terms

Group the terms into pairs to help identify common factors:

(ax + ay) + (bx + by)

Step 2: Factor each group

Find the common factors in each group:

a(x + y) + b(x + y)

Step 3: Find and remove common factors

The expression now reveals a common factor of (x + y) in both groups:

(x + y)(a + b)

Now, the expression has been completely factorized using grouping and common factor extraction.

More examples to reinforce learning

Here are some more examples to further strengthen your understanding of factorization by grouping.

Example 3: Factors

Factor the following expression:

3m^2 + 3mn + 2m + 2n

Solution

  1. Group the terms: (3m^2 + 3mn) + (2m + 2n)
  2. Factor out each group: 3m(m + n) + 2(m + n)
  3. Find the common factors: (m + n)(3m + 2)

Thus, (m + n)(3m + 2) is the factored form.

Example 4: Factors

Consider the polynomial:

pq + pr + qr + q^2

Solution

  1. Group the terms: (pq + pr) + (qr + q^2)
  2. Factor each group: p(q + r) + q(r + q)
  3. Note that (q + r) and (r + q) are the same, so: (q + r)(p + q)

Or in simplified form (q + r)(p + q) This is the factored form.

Identifying when to use factorization by grouping

Factoring by grouping is often best used for polynomials that:

  • There are four positions, which can easily be grouped into pairs.
  • Display symmetry or recurring patterns.
  • This problem does not seem to be solvable using simple factorization methods such as directly taking out a common factor.

For example, if a polynomial has no obvious common factor for all of its terms, but it seems to break down into smaller expressions, then grouping may be the ideal strategy.

Practice problems

Try using factorization by grouping to solve the following problems:

Problem 1

Factor the expression:

2x^2 + 4x + 3x + 6

Problem 2

Factor the expression:

3ab + 3bc + a^2 + ac

Remember to follow the outlined procedures of grouping, factoring, taking out common factors, and simplifying to find your solutions.

Conclusion

Factoring by grouping is a powerful tool in algebra that helps simplify complex expressions. By strategically grouping and factoring terms, students can often transform complex polynomials into more manageable forms. This technique requires careful attention to detail and practice in identifying potential groupings. With practice, recognizing when and how to use factoring by grouping becomes second nature.

Continue to practice with different problems and always check your final factorized form to ensure accuracy. Mastering this technique not only helps in solving algebraic expressions but also strengthens overall problem-solving skills.


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Factorization by Grouping

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