Question

you have to factorize it and solve it.
I will mark brainliast to the correct answer.​

Answer 1

Question :- Factorize :

$\rm \: ab( {x}^{2} + {y}^{2}) + xy( {a}^{2} + {b}^{2})$

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: ab( {x}^{2} + {y}^{2}) + xy( {a}^{2} + {b}^{2})$

$\rm \: = \: ab {x}^{2} + ab {y}^{2} + xy {a}^{2} + xy {b}^{2}$

can be re-arranged as

$\rm \: = \: ab {x}^{2} + xy {a}^{2} + xy {b}^{2}+ ab {y}^{2}$

can be regrouped as

$\rm \: = \: (ab {x}^{2} + xy {a}^{2}) + (xy {b}^{2}+ ab {y}^{2} )$

$\rm \: = \: ax(bx + ay) +by(bx + ay)$

$\rm \: = \: (bx + ay)(ax + by)$

Hence,

$\rm\implies \:ab( {x}^{2} + {y}^{2}) + xy( {a}^{2} + {b}^{2})\\ \red{\rm \: = \: (bx + ay)(ax + by) }\: \: \: \: \: \: \:$

$\rule{190pt}{2pt}$

Additional information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$

Answer 2

Question :-

➻ Factorize the following :-

• ab ( x² + y² ) + xy ( a² + b² )

_____________________

Answer :-

➳ ab ( x² + y² ) + xy ( a² + b² )

➳ abx² + aby² + xya² + xyb²

➳ abx² + xya² + xyb² + aby²

➳ ( abx² + xya² ) + ( xyb² + aby² )

➳ ax ( bx + ay ) + by ( bx + ay )

➳ ( bx + ay ) ( ax + by )

_____________________