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Chapter Explanation
6

Factorization

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Curriculum

Basic Methods & Linear Identities in Factorization

1. Common Factor (Sum): The most basic form where a common term is factored out.

- $ab + ac = a(b + c)$

Ex: 1) Factorize $2x + 4$

Sol:

$2x+4 = 2x+2(2)$

$\quad \quad \quad \quad \quad \ = 2(x+2)$

Ex: 2) Factorize $5xy+10x$

Sol:

$5xy+10x = 5xy+5(2)(x)$

$\quad \quad \quad \quad \quad \quad \quad \ = 5x(y+2)$

Ex: 3) Factorize $12a^{2}b+15ab^{2}$

Sol:

$12a^{2}b+15ab^{2} = 3\times4\times a^{2}\times b+3\times5\times a\times b^{2}$

$\quad \quad \quad \quad \quad \quad \quad \quad \ \ = 3ab(4a+5b)$

2. Common Factor (Difference): Factorization involving a subtraction within the expression.

- $ab - ac = a(b - c)$

Ex: 1) Factorize $22y-33z$

Sol:

$22y-33z = 11 \times 2 \times y - 11 \times 3 \times z$

$\quad \quad \quad \quad \quad \ \ = 11(2y-33z)$

Ex: 2) Factorize $10x^2 - 18x^3 + 14x^4$

Sol:

$10x^2 - 18x^3 + 14x^4 = 2x^2(5 - 9x + 7x^2)$

3. Factorization by Grouping: Used for four-term expressions by grouping them into pairs.

- $ac + ad + bc + bd = (a + b)(c + d)$

Ex: 1) Factorize $2xy+2y+3x+3$

Sol:

$2xy+2y+3x+3$

$\quad \quad \ = 2y(x+1)+3(x+1)$

$\quad \quad \ = (2y+3)(x+1)$

Ex: 2) Factorize $6xy-4y+6-9x$

Sol:

$6xy-4y+6-9x$

$\quad \quad \ = 6xy-4y-9x+6$

$\quad \quad \ = 2y(3x-2)-3(3x-2)$

$\quad \quad \ = (2y-3)(3x-2)$

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