Homogeneous Polynomial/Examples/Arbitrary Example 1
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Example of Homogeneous Polynomial
The polynomial:
- $\map P {x, y} = x^2 + 3 x y + y^2$
is a homogeneous polynomial in which the degree of each term is $2$.
That this is also a homogeneous expression is demonstrated by replacing $x$ by $k x$ and $y$ by $k y$:
\(\ds \map P {k x, k y}\) | \(=\) | \(\ds k^2 x^2 + 3 k^2 x y + k^2 y^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds k^2 \paren {x^2 + 3 x y + y^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds k^2 \map P {x, y}\) |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): homogeneous
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): homogeneous