Question

(ax + by)² + (bx - ay)²​

Answer 1

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Answer 2

Answer:

Factorization is the decomposition of mathematical objects into the product of smaller or simpler objects

Factorizing helps in finding the roots of the factors

Consider the given equation

$=(a x+b y)^{2}+(b x-a y)^{2}=(ax+by) </p><p>2</p><p> +(bx−ay) </p><p>2</p><p> </p><p></p><p>By applying (a+b)^{2}(a+b) </p><p>2</p><p> and (a-b)^{2}(a−b) </p><p>2</p><p> formula for the above equation </p><p></p><p>We get</p><p></p><p>=\left[a^{2} x^{2}+b^{2} y^{2}+2 a b x y\right]+\left[b^{2} x^{2}+a^{2} y^{2}-2 a b x y\right]=[a </p><p>2</p><p> x </p><p>2</p><p> +b </p><p>2</p><p> y </p><p>2</p><p> +2abxy]+[b </p><p>2</p><p> x </p><p>2</p><p> +a </p><p>2</p><p> y </p><p>2</p><p> −2abxy]</p><p></p><p>Simplify the above equation</p><p></p><p>\begin{gathered}\begin{array}{l}{=a^{2} x^{2}+b^{2} y^{2}+2 a b x y+b^{2} x^{2}+a^{2} y^{2}-2 a b x y} \\ {=a^{2} x^{2}+b^{2} y^{2}+b^{2} x^{2}+a^{2} y^{2}}\end{array}\end{gathered} </p><p>=a </p><p>2</p><p> x </p><p>2</p><p> +b </p><p>2</p><p> y </p><p>2</p><p> +2abxy+b </p><p>2</p><p> x </p><p>2</p><p> +a </p><p>2</p><p> y </p><p>2</p><p> −2abxy</p><p>=a </p><p>2</p><p> x </p><p>2</p><p> +b </p><p>2</p><p> y </p><p>2</p><p> +b </p><p>2</p><p> x </p><p>2</p><p> +a </p><p>2</p><p> y </p><p>2</p><p> </p><p> </p><p> </p><p> </p><p> </p><p></p><p>By taking the common terms</p><p></p><p>We get the above equation as</p><p></p><p>=a^{2}\left(x^{2}+y^{2}\right)+b^{2}\left(x^{2}+y^{2}\right)=a </p><p>2</p><p> (x </p><p>2</p><p> +y </p><p>2</p><p> )+b </p><p>2</p><p> (x </p><p>2</p><p> +y </p><p>2</p><p> )</p><p></p><p>Here \left(x^{2}+y^{2}\right)(x </p><p>2</p><p> +y </p><p>2</p><p> ) is written only once and \left(a^{2}+b^{2}\right)(a </p><p>2</p><p> +b </p><p>2</p><p> ) is combined</p><p></p><p>=\left(x^{2}+y^{2}\right)\left(a^{2}+b^{2}\right)=(x </p><p>2</p><p> +y </p><p>2</p><p> )(a </p><p>2</p><p> +b </p><p>2</p><p> )</p><p>$

Hence the given sum is factorized

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