Regrouping Method of Factorization

Definition:

The regrouping method of factorization involves rearranging or grouping terms in an algebraic expression so that a common factor can be identified and factored out. This method is particularly useful when there is no single common factor in all terms but can be found in pairs or groups of terms.

Steps for the Regrouping Method:

1. Identify terms that can be grouped together based on common factors.
2. Group the terms into pairs or suitable groups.
3. Factor out the common factor from each group.
4. Check for a common binomial factor and factor it out if possible.
5. Rewrite the expression in its factored form.

Examples:

Example 1: Factorizing by Regrouping

Factorize: ax + ay + bx + by

Step 1: Group the terms in pairs:
(ax + ay) + (bx + by)

Step 2: Factor out the common term from each group:
a(x + y) + b(x + y)

Step 3: Identify the common binomial factor (x + y) and factor it out:
(x + y)(a + b)

Thus, the factored form is (x + y)(a + b).

Example 2: Factorizing a Four-Term Expression

Factorize: 6xy – 3x + 4y – 2

Step 1: Group the terms:
(6xy – 3x) + (4y – 2)

Step 2: Factor out the common terms from each group:
3x(2y – 1) + 2(2y – 1)

Step 3: Factor out the common binomial factor (2y – 1):
(2y – 1)(3x + 2)

Thus, the factored form is (2y – 1)(3x + 2).

Example 3: Factorizing a Polynomial

Factorize: x² + 5x + 2x + 10

Step 1: Group the terms:
(x² + 5x) + (2x + 10)

Step 2: Factor out the common terms:
x(x + 5) + 2(x + 5)

Step 3: Factor out the common binomial factor (x + 5):
(x + 5)(x + 2)

Thus, the factored form is (x + 5)(x + 2).

Conclusion:

The Regrouping Method is useful when there is no single common factor for all terms, but grouping them strategically allows us to factor out common binomial expressions. This method is widely used in polynomial factorization and simplifying algebraic expressions.