Question

factorise ( ax + by)^2 + ( bx - ay)^2

Answer 1

(ax + by)² + (bx -ay)²

=a²x² +b²y² +2abxy +b²x² +a²y² -2abxy

=a²x² + b²x² + b²y² +a²y²

=x² (a² + b²) + y²(a²+ b²)

=(a² + b²)(x² + y²)

Answer 2

Answer: the given sum is factorized as $\left(x^{2}+y^{2}\right)\left(a^{2}+b^{2}\right)$.

Step-by-step explanation:

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

$= (a x+b y)^{2}+(b x-a y)^{2}$

By applying $(a+b)^{2}$ and $(a-b)^{2}$ formula  for the above equation  

As we get

$=\left[a^{2} x^{2}+b^{2} y^{2}+2 a b x y\right]+\left[b^{2} x^{2}+a^{2} y^{2}-2 a b x y\right]$

Simplify the above equation

$\begin{array}{l}{=a^{2} x^{2}+b^{2} y^{2}+2 a b x y+b^{2} x^{2}+a^{2} y^{2}-2 a b x y} \\ {=a^{2} x^{2}+b^{2} y^{2}+b^{2} x^{2}+a^{2} y^{2}}\end{array}$

By taking the common terms

We get the above equation as

$=a^{2}\left(x^{2}+y^{2}\right)+b^{2}\left(x^{2}+y^{2}\right)$

Here $\left(x^{2}+y^{2}\right)$ is written only once and $\left(a^{2}+b^{2}\right)$ is combined

$=\left(x^{2}+y^{2}\right)\left(a^{2}+b^{2}\right)$

Hence, the given sum is factorized.

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