Question

if p+q+r = 1 and pq+qr+pr = -1 and pqr = -1, find the value of p³+q³+r³
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Answer 1

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Answer 2

Given:

p+q+r=1

pq+qr+pr = -1

and pqr = -1

To Find:

p³+q³+r³=?

Using Algebraic Identities:

(a+b+c)²=a²+b²+c+2ab+2bc+2ac

a³+b³+c³-3abc=(a+b+c)(a²+b²+c²-ab-bc-ca)

Solution:

(p+q+r)²=p²+q²+r²+2pq+2qr+2pr

(p+q+r)²=p²+q²+r²+2(pq+qr+pr)

(1)²=p²+q²+r²+2(-1)

1=p²+q²+r²-2

p²+q²+r²=3

Now

p³+q³+r³-3pqr=(p+q+r)(p²+q²+r²-pq-qr-pr)

p³+q³+r³=(p+q+r)[p²+q²+r²-(pq+qr+pr)]+3pqr

p³+q³+r³=(1)[3-(-1)]+3(-1)

p³+q³+r³=4-3

p³+q³+r³=1