Question

factorize 27xpower 3+125y^3​

Answer 1

We have been asked to factorize 27x³ + 125y³.

If we carefully observe the given problem ; it is in the form of x³ + y³ = (x+y)³ identity. The expansion :

➝ (x+y)³ = x³ + y³ + 3x²y + 3xy²

For this problem we just need to simplify 27x³ and 125y³.

27x³ can be written as ➝ (3x)³

125y³ can be written as ➝ (5y)³

This is again in the form of :

x³+y³ which is (x+y)³ = (3x+5y)³

So the factorization is :

(3x+5y)³ or (3x+5y)(3x+5y)(3x+5y)

Answer 2

$\large\bf\underline{Answer :-}$

$\bf\large(3x \: + \: 5y \: )( 9x^{2} - 15xy + 25y^{2} )$

$\bf\large\underline\pink{Step\: by \:step\: explaination\: :-}$

$\large\bf\red {It \:can \: be \: written \: as \:} :- \\ {27} \: = {3}^{3} \\ 125 = {5}^{3}$

SO,

$\large \bf {(3x)}^{3} + {(5y)}^{3}$

$\large \bf \red\: {a}^{3} + {b}^{3} \implies \: (a + b)( {a}^{2} - ab + {b}^{2}) \\\bf\large\blue{ a \: \implies \: 3x }\\ \:\bf\large\blue{ b\implies \: 5y}$

$\bf{\green{Putting\:the\:values}} \\ \\ \\ :\implies \bf{(3x + 5y)[(3x)^{2}] - (3x)(5y) + [(5y)^{2}]} \\ \\ \\\large{\bf{\orange{Simplify :-}}} \\ \\ \\ :\implies \large{\bf{(3x + 5y)(9x^{2} - 15xy + 25y^{2})}}$

Let's explore more :-

Some algebraic identities are as follows :-

$\bf\purple{(a + b)^{2} \implies \: {a}^{2} + 2ab + {b}^{2} }$

$\bf\orange{(a - b)^{2} \implies \: {a}^{2} - 2ab + {b}^{2} }$

$\bf\green{(a + b)(a - b) \implies \: {a}^{2} - {b}^{2} }$

$\bf\orange{(x + a)(x \: + b) \implies \: {x}^{2} + (a + b)x + ab}$

$\bf\purple{(a + b)^{3} \implies \: {a}^{3} + 3 {a}^{2} b +3a {b}^{2} + {b}^{3} }$

$\bf\green{(a - b)^{3} \implies \: {a}^{3} - 3 {a}^{2} b +3a {b}^{2} - {b}^{3}}$