Q. 3 E5.0( 1 Vote )

Factorize the following expressions:

25x2 + 4y2 + 9z2 – 20xy + 12yz – 30zx

Answer :


Method 1

The above equation can be simplified as :
= (– 5x)2 + (2y)2 + (3z)2 + 2(– 5x)(2y) + 2(2y)(3z) + 2(3z)(– 5x)
This equation is of the form: p2 + q2 + r2 + 2pq + 2qr + 2rp


where p = – 5x, q = 2y, r = 3z
Using the identity: p2 + q2 + r2 + 2pq + 2qr + 2rp = (p + q + r)2
We get,
25x2 + 4y2 + 9z2 – 20xy + 12yz – 30zx = (– 5x + 2y + 3z)2


= (– 5x + 2y + 3z)2


= (– 1)2(5x – 2y – 3z)2


= (5x – 2y – 3z)2


Method 2


The above equation can be simplified as:


= (– 5x)2 + (2y)2 + (3z)2 + 2(– 5x)(2y) + 2(2y)(3z) + 2(3z)(– 5x)
= {
(– 5x)2 + 2(– 5x)(2y) + (2y)2} + (3z)2 + 2(2y)(3z) + 2(3z)(– 5x)
(
p2 + 2pq + q2 = (p + q)2)
= (– 5x + 2y)2 + (3z)2 + 2(2y)(3z) + 2(3z)(– 5x)
Taking 2(3z) common in term 2(2y)(3z) + 2(3z)(– 5x)
= (– 5x + 2y)2 + (3z)2 + 2(3z)(2y – 5x)
= (– 5x + 2y)2 + 2(3z)(2y – 5x) + (3z)2


This of the form: (p + q)2 + 2r(p + q) + r2


Where, p = – 5x, q = 2y and r = 3z


Using the identity: (p + q)2 = p2 + 2pq + q2


We get,


25x2 + 4y2 + 9z2 – 20xy + 12yz – 30zx = (– 5x + 2y + 3z)2


= (– 5x + 2y + 3z)2


= (– 1)2(5x – 2y – 3z)2


= (5x – 2y – 3z)2


25x2 + 4y2 + 9z2 – 20xy + 12yz – 30zx = (5x – 2y – 3z)2


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