Each of x, y, z is a distinct real number and n is a positive integer such that:
x2+y2+z2-xy-yz-zx is a factor of (x-y)n+(y-z)n+(z-x)n
Find the form of n.
<h3 data-start="254" data-end="299">Simplify the Quadratic Expression
Escape Road</h3>
First, rewrite the quadratic expression:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mi>y</mi><mo>−</mo><mi>y</mi><mi>z</mi><mo>−</mo><mi>z</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">x^2 + y^2 + z^2 - xy - yz - zx</annotation></semantics></math>x2+y2+z2−xy−yz−zx
This is a well-known symmetric polynomial, which can be factored as:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo fence="true">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>z</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\frac{1}{2} \left( (x-y)^2 + (y-z)^2 + (z-x)^2 \right)</annotation></semantics></math>21((x−y)2+(y−z)2+(z−x)2)
good
