Factorize the following expressions:
p2 + q2 + r2 + 2pq + 2qr + 2rp
Method 1
= p2 + q2 + r2 + 2pq + 2qr + 2rp
= (p)2 + (q)2 + (r)2 + 2(p)(q) + 2(q)(r) + 2(r)(p)
It is of the from a2 + b2 + c2 + 2ab + 2bc + 2ca
where a = p, b = q, c = r
Using the identity (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
We get,
⇒ p2 + q2 + r2 + 2pq + 2qr + 2rp = (p + q + r)2
Method 2
= p2 + q2 + r2 + 2pq + 2qr + 2rp
The above equation can be simplified as
= p2 + 2pq + q2 + r2 + 2qr + 2rp
= (p + q)2 + r2 + 2qr + 2rp (∵ a2 + 2ab + b2 = (a + b)2)
Taking 2r common in the term 2qr + 2rp
we get,
= (p + q)2 + r2 + 2r(q + p)
Rearranging the above equation as
= (p + q)2 + 2r(q + p) + r2
This is of the from a2 + 2ab + b2
Where, a = p + q and b = r
Using the identity: (a + b)2 = a2 + 2ab + b2
We get,
⇒ p2 + q2 + r2 + 2pq + 2qr + 2rp = ((p + q) + r)2
= (p + q + r)2
p2 + q2 + r2 + 2pq + 2qr + 2rp = (p + q + r)2
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