Factorization in Mathematics Explained in Detailed

In This Chapter We cover the all the topics and basics of factorization these are the topics we cover as follows:

Highest Common Factor (H.C.F) of Monomials

The H.C.F of given monomials is the common factor having greatest coefficient and highest powers of the variables.

Ex: 1) Find H.C.F of $6x^{3}y$ and $18x^{2}y^{3}$.

Sol: $6x^{3}y = 2 \times 3 \times x^{3} \times y$

$\quad \quad \ 18x^{2}y^{3} = 2 \times 3^{2} \times x^{2} \times y^{3}$

$\quad \quad \ \text{H.C.F} = 2 \times 3 \times x^{2} \times y$

$\quad \quad \quad \quad \quad \quad \ = 6x^{2}y$

Ex: 2) Find H.C.F of $5xy$ and $10x$.

Sol: $5xy = 5 \times x \times y$

$\quad \quad \ 10x = 2 \times 5 \times x$

$\quad \quad \ \text{H.C.F} = 5x$

Ex: 3) H.C.F of $12a^{2}b$ and $15ab^{2}$ is 3ab

Ex: 4) H.C.F of $2x$ and $4$ is 2

Ex: 5) H.C.F of $12x$ and $36$ is 12

Ex: 6) H.C.F of $14pq$ and $35pqr$ is 7pq

What is Factorization and its identities?

Factorization is the mathematical process of expressing an algebraic expression as a product of two or more factors. Below are the 15 essential identities and methods used to break down expressions:

Basic Methods & Linear Identities

1. $ab + ac = a(b + c)$
2. $ab - ac = a(b - c)$
3. $ac + ad + bc + bd = (a + b)(c + d)$

Square-Based Identities

4. $a^2 - b^2 = (a + b)(a - b)$

5. $a^2 + 2ab + b^2 = (a + b)^2$

6. $a^2 - 2ab + b^2 = (a - b)^2$

7. $x^2 + (a + b)x + ab = (x + a)(x + b)$

8. $a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a + b + c)^2$

9. $a^2 + b^2 + c^2 + 2ab - 2bc - 2ca = (a + b - c)^2$

Cubic & Advanced Identities

10. $a^3 + 3a^2b + 3ab^2 + b^3 = (a + b)^3$

11. $a^3 - 3a^2b + 3ab^2 - b^3 = (a - b)^3$

12. $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

13. $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

14. $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$

15. If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$

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:

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

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

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

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

Square Based Identities in Factorization

4. Difference of Two Squares

$$a^2 - b^2 = (a + b)(a - b)$$

Ex 1: $49y^2 - 36 = (7y)^2 - (6)^2 = \mathbf{(7y + 6)(7y - 6)}$

Ex 2: $p^4 - 81 = (p^2 + 9)(p^2 - 9) = \mathbf{(p^2 + 9)(p - 3)(p + 3)}$

Ex 3 (Mixed): $a^2 - 2ab + b^2 - c^2 = (a - b)^2 - c^2 = \mathbf{(a - b + c)(a - b - c)}$

5. Perfect Square Trinomials

These are the results of squaring a binomial sum or difference.

Sum Identity:

$a^2 + 2ab + b^2 = (a + b)^2$

Ex: $x^2 + 8x + 16 = \mathbf{(x + 4)^2}$

Ex: $25m^2 + 30m + 9 = \mathbf{(5m + 3)^2}$

Difference Identity:

$a^2 - 2ab + b^2 = (a - b)^2$

Ex: $4y^2 - 12y + 9 = \mathbf{(2y - 3)^2}$

6. Perfect Square Trinomial (Difference): The result of squaring a binomial difference.

$a^2 - 2ab + b^2 = (a - b)^2$

Ex 1: Factorize $4y^2 - 12y + 9$

Sol:

$(2y)^2 - 2(2y)(3) + (3)^2 = \mathbf{(2y - 3)^2}$

Ex 2: Factorize $a^2 - 2ab + b^2 - c^2$

Sol:

$(a - b)^2 - c^2 = [(a - b) + c][(a - b) - c] = \mathbf{(a - b + c)(a - b - c)}$

7. Trinomials with Constant Factors (Middle Term Splitting)

$$x^2 + (a + b)x + ab = (x + a)(x + b)$$

Ex 1:

$x^2 + 5x + 6 = \mathbf{(x + 3)(x + 2)}$

Ex 2:

$y^2 - 7y + 12 = \mathbf{(y - 4)(y - 3)}$

Ex 3

($ax^2 + bx + c$): $3x^2 + 10x + 3 = \mathbf{(x + 3)(3x + 1)}$

Ex 4 (Advanced):

$6x^2 + 5x - 6 = \mathbf{(2x + 3)(3x - 2)}$

Ex 5 (Substitution):

$(2a - b)^2 + 2(2a - b) - 8 = \mathbf{(2a - b - 2)(2a - b + 4)}$

8. Square of a Trinomial Standard Identity

Factoring expressions with three variables ($a, b, \text{ and } c$).

Standard Identity:

$a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a + b + c)^2$

Ex 1:

$4a^2 + b^2 + c^2 + 4ab + 2bc + 4ac = \mathbf{(2a + b + c)^2}$

Ex 2:

$9a^2 + 4b^2 + 16 + 12ab + 16b + 24a = \mathbf{(3a + 2b + 4)^2}$

9. Square of a Trinomial Variable Signs Identity

$a^2 + b^2 + c^2 + 2ab - 2bc - 2ca = (a + b - c)^2$

• Usage: Used when the signs of the product terms involving one specific variable are negative.

Ex: Factorize $25x^2 + y^2 + 4z^2 - 10xy - 4yz + 20zx$

Sol:

$25x^2 + y^2 + 4z^2 - 10xy - 4yz + 20zx$

$\quad \quad = (5x)^2 + (-y)^2 + (2z)^2 + 2(5x)(-y) + 2(-y)(2z) + 2(2z)(5x)$

$\quad \quad = \mathbf{(5x - y + 2z)^2}$

Cubic & Advanced Identities in Factorization

10. Cube of a Binomial (Sum): Factoring a four-term perfect cubic expression.

$a^3 + 3a^2b + 3ab^2 + b^3 = (a + b)^3$

Ex: 1) Factorize $8a^3 + b^3 + 12a^2b + 6ab^2$

Sol:

$8a^3 + b^3 + 12a^2b + 6ab^2$

$\quad \quad = (2a)^3 + (b)^3 + 3(2a)^2(b) + 3(2a)(b)^2$

$\quad \quad = \mathbf{(2a + b)^3}$

11. Cube of a Binomial (Difference): The cubic version of the subtraction identity.

$a^3 - 3a^2b + 3ab^2 - b^3 = (a - b)^3$

Ex: 1) Factorize $27 - 125a^3 - 135a + 225a^2$

Sol:

$27 - 125a^3 - 135a + 225a^2$

$\quad \quad = (3)^3 - (5a)^3 - 3(3)^2(5a) + 3(3)(5a)^2$

$\quad \quad = \mathbf{(3 - 5 a)^3}$

12. Sum of Two Cubes: Breaking down the sum of two cubic terms.

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Ex: 1) Factorize $y^3 + 125$

Sol:

$y^3 + 125$ $\quad \quad$

$= (y)^3 + (5)^3$ $\quad \quad$

$= (y+5)(y^2 - y(5) + (5)^2)$ $\quad \quad$

$= \mathbf{(y+5)(y^2 - 5y + 25)}$

13. Difference of Two Cubes: Breaking down the subtraction of two cubic terms.

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Ex: 1) Factorize $\frac{x^3}{216} - 8y^3$

Sol:

$\frac{x^3}{216} - 8y^3$

$= \left(\frac{x}{6}\right)^3 - (2y)^3$ $\quad \quad \quad \quad \quad \$

$= \left(\frac{x}{6} - 2y\right) \left[ \left(\frac{x}{6}\right)^2 + \frac{x}{6}(2y) + (2y)^2 \right]$ $\quad \quad \quad \quad \quad \$

$= \mathbf{\left(\frac{x}{6} - 2y\right) \left( \frac{x^2}{36} + \frac{xy}{3} + 4y^2 \right)}$

14. The Long Cubic Identity: A complex identity involving three variables.

$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$

Ex: 1) Factorize $a^{3}-b^{3}+1+3ab$

Sol: $a^{3}-b^{3}+1+3ab$

$\quad \quad = (a)^{3}+(-b)^{3}+(1)^{3}-3(a)(-b)(1)$

$\quad \quad = (a-b+1)(a^{2}+(-b)^{2}+(1)^{2}-a(-b)-(-b)(1)-(1)(a))$

$\quad \quad = \mathbf{(a-b+1)(a^{2}+b^{2}+1+ab+b-a)}$

15. Conditional Cubic Property: A special rule applied when the sum of the variables is zero.

If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$

Ex: 1) Factorize $(x-y)^{3}+(y-z)^{3}+(z-x)^{3}$

Sol: $(x-y)^{3}+(y-z)^{3}+(z-x)^{3}$

$\quad \quad = a^{3}+b^{3}+c^{3} \text{ where } a+b+c = x-y+y-z+z-x = 0$

$\quad \quad = 3abc$

$\quad \quad = \mathbf{3(x-y)(y-z)(z-x)}$

Division of Algebraic Expressions

Division of a monomial by another monomial by using factorisation

Ex: 1) Divide $6x^3$ by $2x$

Sol:

$\frac{6x^3}{2x} = 3x^2$

2) Divide $-20x^4$ by $10x^2$

Sol:

$\frac{-20x^4}{10x^2} = -2x^2$

3) Divide $7x^2y^2z^2$ by $14xyz$

Sol:

$\frac{7x^2y^2z^2}{14xyz} = \frac{xyz}{2}$

Division of Polynomial by a monomial by using factorisation

Ex: 1) Divide $24(x^2yz + xy^2z + xyz^2)$ by $8xyz$

Sol:

$\frac{24(x^2yz + xy^2z + xyz^2)}{8xyz}$

$\quad \quad = \frac{24xyz(x+y+z)}{8xyz}$

$\quad \quad = \mathbf{3(x+y+z)}$

Ex: 2) Divide $3y^8 - 4y^6 + 5y^4$ by $y^4$

Sol:

$\frac{3y^8 - 4y^6 + 5y^4}{y^4}$

$\quad \quad = \frac{{y^4}(3y^4 - 4y^2 + 5)}{{y^4}}$

$\quad \quad = \mathbf{3y^4 - 4y^2 + 5}$