Example 1: Trinomial Square

Expression to Factorize: $25x^2 + y^2 + 4z^2 – 10xy – 4yz + 20zx$

Solution:

1. Identify the squared terms ($A^2, B^2, C^2$):

   • $25x^2 = (5x)^2$

   • $y^2 = (-y)^2$ or $(y)^2$

   • $4z^2 = (2z)^2$

2. Determine the signs using the middle terms:

   • The term $-10xy$ indicates a negative sign for either $5x$ or $y$.

   • The term $-4yz$ indicates a negative sign for either $y$ or $2z$.

   • The term $+20zx$ indicates $5x$ and $2z$ have the same sign.

   • The common negative term must be $y$. Let $A=5x$, $B=-y$, and $C=2z$.

3. Rewrite the expression in the identity form $A^2 + B^2 + C^2 + 2AB + 2BC + 2CA$:
   $$(5x)^2 + (-y)^2 + (2z)^2 + 2(5x)(-y) + 2(-y)(2z) + 2(2z)(5x)$$
4. Apply the identity $(A+B+C)^2$:
   $$(5x – y + 2z)^2$$

II. Factorization using $(a \pm b)^3$ (Cubic Identities)

Here, we use the identities for the cube of a binomial:

• Note 1: $a^3 + 3a^2b + 3ab^2 + b^3 = (a+b)^3$

• Note 2: $a^3 – 3a^2b + 3ab^2 – b^3 = (a-b)^3$

Example 1: Factorizing $(a+b)^3$

Expression to Factorize: $8a^3 + b^3 + 12a^2b + 6ab^2$

Solution:

1. Identify the cube terms ($a^3$ and $b^3$):

   • $8a^3 = (2a)^3$

   • $b^3 = (b)^3$

2. Rewrite the expression using the identity pattern:
   $$(2a)^3 + (b)^3 + 3(2a)^2(b) + 3(2a)(b)^2$$
3. Apply the identity $(a+b)^3$:
   $$(2a + b)^3$$

Example 2: Factorizing $(a-b)^3$

Expression to Factorize: $27 – 125a^3 – 135a + 225a^2$

Solution:

1. Rearrange the terms into the identity order $a^3 + 3a^2b + 3ab^2 + b^3$ (or $a^3 – b^3$ etc.):
   $$27 – 135a + 225a^2 – 125a^3$$

2. Identify the cube terms ($a^3$ and $b^3$):

   • $27 = (3)^3$

   • $125a^3 = (5a)^3$

3. Rewrite the expression using the identity pattern ($a^3 – 3a^2b + 3ab^2 – b^3$):
   $$(3)^3 – 3(3)^2(5a) + 3(3)(5a)^2 – (5a)^3$$
4. Apply the identity $(a-b)^3$:
   $$(3 – 5a)^3$$

III. Factorization using $a^3 \pm b^3$ (Sum and Difference of Cubes)

These identities are used to factorize the sum or difference of two perfect cubes:

• Note 1 (Sum of Cubes): $a^3 + b^3 = (a+b)(a^2 – ab + b^2)$

• Note 2 (Difference of Cubes): $a^3 – b^3 = (a-b)(a^2 + ab + b^2)$

Example 1: Sum of Cubes

Expression to Factorize: $y^3 + 125$

Solution:

1. Identify the cube terms ($a^3$ and $b^3$):

   • $y^3 = (y)^3$

   • $125 = (5)^3$

2. Apply the Sum of Cubes identity $a^3 + b^3 = (a+b)(a^2 – ab + b^2)$:
   $$(y + 5)((y)^2 – (y)(5) + (5)^2)$$
3. Simplify:
   $$(y + 5)(y^2 – 5y + 25)$$

Example 2: Difference of Cubes

Expression to Factorize: $\frac{x^3}{216} – 8y^3$

Solution:

1. Identify the cube terms ($a^3$ and $b^3$):

   • $\frac{x^3}{216} = \left(\frac{x}{6}\right)^3$ (Since $6^3 = 216$)

   • $8y^3 = (2y)^3$

2. Apply the Difference of Cubes identity $a^3 – b^3 = (a-b)(a^2 + ab + b^2)$:
   $$\left(\frac{x}{6} – 2y\right) \left(\left(\frac{x}{6}\right)^2 + \left(\frac{x}{6}\right)(2y) + (2y)^2\right)$$
3. Simplify:
   $$\left(\frac{x}{6} – 2y\right) \left(\frac{x^2}{36} + \frac{2xy}{6} + 4y^2\right)$$
4. Further Simplify the fraction $\frac{2xy}{6}$ to $\frac{xy}{3}$:
   $$left(frac{x}{6} – 2yright) left(frac{x^2}{36} + frac{xy}{3} + 4y^2right)$