Question

Factorise the following
${x}^4 - 2 {x }^{2} {y}^{2} + {y}^4 \\ 4(a + b {)}^{2} - 9(a - b {)}^{2}$
​

Answer 1

Answer :-

a) $\sf {x}^4 - 2 {x}^{2} {y}^{2} + {y}^4$

$\implies\sf (x^2)^2 - 2(x^2)(y^2) + (y^2)^2$

• $\sf a^2 - 2ab + b^2 = ( a - b)^2$

$\implies\sf (x^2 - y^2)^2$

$\implies\sf (x^2 - y^2)(x^2 - y^2)$

$\implies\sf [(x)^2 - (y)^2][(x)^2 - (y)^2]$

• $\sf a^2 - b^2 = (a+b)(a-b)$

$\implies\sf (x + y)(x - y)(x+y)(x-y)$

$\boxed{\sf {x}^4 - 2 {x}^{2} {y}^{2} + {y}^4 = (x + y)(x - y)(x+y)(x-y)}$

b) $\sf 4(a + b)^{2} - 9(a - b)^{2}$

$\implies\sf [2(a+b)]^2 - [3(a-b)^2]$

$\implies\sf [2a + 2b]^2 - [3a - 3b]^2$

• $\sf a^2 - b^2 = (a+b)(a-b)$

$\implies\sf (2a + 2b + 3a - 3b)(2a + 2b - 3a + 3b)$

$\implies\sf (5a - b)(5b - a)$

$\boxed{\sf 4(a + b)^{2} - 9(a - b)^{2} = (5a - b)(5b - a)}$

Additional information :-

$\boxed{\bigstar\:\:\textbf{\textsf{Algebraic\:Identity}}\:\bigstar}\\\\1)\sf\:(A+B)^{2} = A^{2} + 2AB + B^{2}\\\\2)\sf\: (A-B)^{2} = A^{2} - 2AB + B^{2}\\\\3)\sf\: A^{2} - B^{2} = (A+B)(A-B)\\\\4)\sf\: (A+B)^{2} = (A-B)^{2} + 4AB\\\\5)\sf\: (A-B)^{2} = (A+B)^{2} - 4AB\\\\6)\sf\: (A+B)^{3} = A^{3} + 3AB(A+B) + B^{3}\\\\7)\sf\:(A-B)^{3} = A^{3} - 3AB(A-B) + B^{3}\\\\8)\sf\: A^{3} + B^{3} = (A+B)(A^{2} - AB + B^{2})$