Homogeneous Polynomial/Examples/Arbitrary Example 1

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