x^3-(p+q-r)x^2+(pq-pr-qr)x+pqr
How to factor this??
x^3-(p+q-r)x^2+(pq-pr-qr)x+pqr
If it factors, the p, or q, or r must be solutions to
x^3-(p+q-r)x^2+(pq-pr-qr)x+pqr = 0
let f(x) = x^3-(p+q-r)x^2+(pq-pr-qr)x+pqr
try x = p
f(p) = p^3 -p^2(p+q-r) + (pq - pr - qr)(p) + pqr
= p^3 - p^3 - qp^2 + rp^2 + qp^2 - rp^2 - prq + pqr
= 0
so (x-p) is a factor
in the same way, I can show that f(q) = 0
so x-q is a factor
leaving me with
(x-p)(x-q)(x .... r) = 0
realizing where the first and last terms come from
x^3 comes from (x...)(x...)(x....)
and pqr comes from (.. -p)(.. -q)(.. ?? r)
so it must be +r
x^3-(p+q-r)x^2+(pq-pr-qr)x+pqr
=(x-p)(x-q)(x+r)
also check out the way the coefficients are sums of subsets of the roots:
http://mathforum.org/library/drmath/view/61024.html
To factor the given expression, x^3 - (p+q-r)x^2 + (pq-pr-qr)x + pqr, we can follow these steps:
Step 1: Identify the common factors, if any. In this case, there aren't any common factors to factor out.
Step 2: Look for any recognizable patterns. In this case, we do not see any recognizable patterns.
Step 3: Apply the grouping method. We group the terms in pairs and look for a common factor within each group.
(x^3 - (p+q-r)x^2) + ((pq-pr-qr)x + pqr)
We can factor out an x^2 from the first group and pq from the second group:
x^2 (x - (p+q-r)) + pq (x - (p+q-r))
Step 4: Notice that we have a common factor, (x - (p+q-r)), we can factor that out:
(x - (p+q-r))(x^2 + pq)
Therefore, the factored expression is: (x - (p+q-r))(x^2 + pq)