Problem 8 · Link symmetric expressions through identities · 5 points
Made by AI and not reviewed by a real person.
Three real numbers x, y, and z satisfy x+y+z=9 and (x−y)²+(y−z)²+(z−x)²=18. What is (x+y)(y+z)(z+x)+xyz?
- 162
- 200
- 189
- 243
- 216
Show a hint
- The clues don't determine each number, but they might determine a symmetric combination that appears in the expression you want.
- Let P=xy+yz+zx. Rewrite the squared differences using P, then expand the product and see what happens to xyz.
Show the answer and solution
Answer: E — 216
Let S=x+y+z=9 and P=xy+yz+zx. Expanding gives (x−y)²+(y−z)²+(z−x)²=2(S²−3P), so 18=2(81−3P) and P=24. Also, (x+y)(y+z)(z+x)+xyz=SP, giving 9·24=216. The tempting 189 comes from missing the factor of 2 in the squared-differences identity.