if the zeroes of the polynomial f(x) = x3-3x3-6x+8 are (p-q),p and (p+q), then the value of p and q is?
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Let the zeros of the polynomial be :
a-d, a and a+d, so that the roots are in AP.
f(x)=x³+3px²+3qx+r.
The standard form of a cubic equation is:
x³-(a+b+c)x²+(ab+bc+ca)x-abc=0.
Comparing this equation with the given polynomial:
We find:
3p=-(a-d+a+a+d).
=>3p=-3a
=>p=-a
3q=(a-d)a+a(a+d)+(a+d)(a-d)
=>3q=a²-ad+a²+ad+a²-d².
=>3q=3a²-d².
=> d²=3a²-3q
Or, d²=3p²-3q
And, r=-(a-d)a(a+d)
Or, r=ad²-a³
Or, r=(-p)(3p²-3q)-(-p)³
Or, r=-3p³+3pq+p³
Thus, r=3pq-2p³.
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a-d, a and a+d, so that the roots are in AP.
f(x)=x³+3px²+3qx+r.
The standard form of a cubic equation is:
x³-(a+b+c)x²+(ab+bc+ca)x-abc=0.
Comparing this equation with the given polynomial:
We find:
3p=-(a-d+a+a+d).
=>3p=-3a
=>p=-a
3q=(a-d)a+a(a+d)+(a+d)(a-d)
=>3q=a²-ad+a²+ad+a²-d².
=>3q=3a²-d².
=> d²=3a²-3q
Or, d²=3p²-3q
And, r=-(a-d)a(a+d)
Or, r=ad²-a³
Or, r=(-p)(3p²-3q)-(-p)³
Or, r=-3p³+3pq+p³
Thus, r=3pq-2p³.
4.4k Views · View Upvoters · Answer requested by Sonal Panjwani
Your response is private.
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