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6

Factorization

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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:

$y^3 + 125$ $\quad \quad = (y)^3 + (5)^3$

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

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

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

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:

$\frac{x^3}{216} - 8y^3 = \left(\frac{x}{6}\right)^3 - (2y)^3$

$\frac{x^3}{216} - 8y^3 = \left(\frac{x}{6}\right)^3 - (2y)^3$ $\quad \quad \quad \quad \quad \ \ = \left(\frac{x}{6} - 2y\right) \left[ \left(\frac{x}{6}\right)^2 + \frac{x}{6}(2y) + (2y)^2 \right]$

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

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

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

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

$\quad \quad = 3abc$

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

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