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If x + y + z = 10 and xy + yz + zx = 25, then what is the value of x³ + y³ + z³ - 3xyz?
250
275
300
325
250
Notice that if x=5,y=5,z=0, then x+y+z=10 and xy+yz+zx=25+0+0=25. Thus, the expression becomes 53+53+03−3(5)(5)(0)=125+125=250.
x + y + z = 10 and xy + yz + zx = 25
x3+y3+z3−3xyz=(x+y+z)(x2+y2+z2−(xy+yz+zx))
Notice that if x=5,y=5,z=0, then x+y+z=10 and xy+yz+zx=25+0+0=25. Thus, the expression becomes 53+53+03−3(5)(5)(0)=125+125=250.
Many students forget to square the sum (x+y+z)2 to find (x2+y2+z2) and instead mistakenly equate (x2+y2+z2) directly to (x+y+z)2.
Calculate sum of squares
Use the identity (x+y+z)2=x2+y2+z2+2(xy+yz+zx) to find x2+y2+z2. Given x+y+z=10 and xy+yz+zx=25, we get 102=x2+y2+z2+2(25).
x2+y2+z2=100−50=50
Apply the cubic identity
Substitute the known values into the identity x3+y3+z3−3xyz=(x+y+z)(x2+y2+z2−(xy+yz+zx)).
(10)(50−25)
Final calculation
Compute the product of the terms derived in the previous steps to find the final result.
10×25=250
A is correct because the substitution of the given values into the identity results in exactly 250.
This identity is a fundamental building block in solving polynomial systems in higher algebra and is frequently tested in competitive exams like SSC CGL and banking aptitude sections.
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