Question

Factorise the following
$\large \red{\underbrace{\bf{ \pink{ {a}^{4} - ({a}^{2} - 5p^{2})^{2} } }}}$
​

Answer 1

$\rm a^{4}-(a^{2}-5p^{2})^{2}$

$\;$

$\rm=(a^{2})^{2}-(a^{2}-5p^{2})^{2}$

$\;$

$\boxed{\rm A^{2}-B^{2}=(A+B)(A-B)}$

$\rm=(a^{2}+a^{2}-5p^{2})(a^{2}-a^{2}+5p^{2})$

$\;$

$\rm=5p^{2}(2a^{2}-5p^{2})$

$\;$

$\rm\therefore a^{4}-(a^{2}-5p^{2})^{2}=5p^{2}(2a^{2}-5p^{2})$

$\;$

$\Large\bigstar\textrm{Note}$

In your previous question, check if the polynomial is correct.

Another answer has a mistake. I'm telling it for you. See the box carefully.

$\sf2x^{3}-4x^{2}+3x-6-2xy+y$

$\sf=(2x^{3}-4x^{2})+(3x-6)-(\boxed{\sf2xy-y})$

$\sf=(2x^{3}-4x^{2})+(3x-6)-\boxed{\sf2y(x-2)}$

$\;$

The boxed term becomes,

$\sf 2y(x-2)=2xy-4y$

$\;$

$\sf 2xy-y$ and $\sf 2xy-4y$ are different.

Answer 2

$\rule{300pt}{0.1pt}$

$\rm a^{4}-(a^{2}-5p^{2})^{2}$

$\rm=(a^{2})^{2}-(a^{2}-5p^{2})^{2}$

$\rm=(a^{2}+a^{2}-5p^{2})(a^{2}-a^{2}+5p^{2})</p><p>\;$

$\rm=5p^{2}(2a^{2}-5p^{2})=5p$

$\rm\therefore a^{4}-(a^{2}-5p^{2})^{2}=5p^{2}(2a^2 - 5 {p}^{2} )$