Answer:
one book, one pen, one techer and one student can change the world.
Step-by-step explanation:
mark as brainliest.
[tex]\large\underline{\sf{Solution-}}[/tex]
Given expression is
[tex]\sf \: ab( {x}^{2} + {y}^{2} ) - xy( {a}^{2} + {b}^{2}) \\ \\ [/tex]
[tex]\sf \: = \: ab {x}^{2} + ab{y}^{2} - xy {a}^{2} - xy{b}^{2} \\ \\ [/tex]
can be re-arranged as
[tex]\sf \: = \: ab {x}^{2} - xy {a}^{2} + ab {y}^{2} - xy{b}^{2} \\ \\ [/tex]
can be regrouped as
[tex]\sf \: = \: (ab {x}^{2} - xy {a}^{2}) + (ab {y}^{2} - xy{b}^{2}) \\ \\ [/tex]
[tex]\sf \: = \: ax(bx - ay) + by(ay - bx) \\ \\ [/tex]
[tex]\sf \: = \: ax(bx - ay) - by(bx - ay) \\ \\ [/tex]
[tex]\sf \: = \: (bx - ay) \: (ax - by) \\ \\ [/tex]
Hence,
[tex]\sf \: \bf\implies \:ab( {x}^{2} + {y}^{2} ) - xy( {a}^{2} + {b}^{2}) = \: (bx - ay) \: (ax - by) \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information :-
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{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}[/tex]
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Answers & Comments
Answer:
(ay - bx)(by - ax)
one book, one pen, one techer and one student can change the world.
Step-by-step explanation:
mark as brainliest.
[tex]\large\underline{\sf{Solution-}}[/tex]
Given expression is
[tex]\sf \: ab( {x}^{2} + {y}^{2} ) - xy( {a}^{2} + {b}^{2}) \\ \\ [/tex]
[tex]\sf \: = \: ab {x}^{2} + ab{y}^{2} - xy {a}^{2} - xy{b}^{2} \\ \\ [/tex]
can be re-arranged as
[tex]\sf \: = \: ab {x}^{2} - xy {a}^{2} + ab {y}^{2} - xy{b}^{2} \\ \\ [/tex]
can be regrouped as
[tex]\sf \: = \: (ab {x}^{2} - xy {a}^{2}) + (ab {y}^{2} - xy{b}^{2}) \\ \\ [/tex]
[tex]\sf \: = \: ax(bx - ay) + by(ay - bx) \\ \\ [/tex]
[tex]\sf \: = \: ax(bx - ay) - by(bx - ay) \\ \\ [/tex]
[tex]\sf \: = \: (bx - ay) \: (ax - by) \\ \\ [/tex]
Hence,
[tex]\sf \: \bf\implies \:ab( {x}^{2} + {y}^{2} ) - xy( {a}^{2} + {b}^{2}) = \: (bx - ay) \: (ax - by) \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information :-
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{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}[/tex]