What is homogeneous function ofa degree in a polynomial? Explain.

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{A} \mathrm{homogeneous} \mathrm{polynomial} \mathrm{is} \mathrm{a} \mathrm{polynomial} \mathrm{whose} \mathrm{nonzero} \mathrm{terms} \mathrm{all} \mathrm{have} \\ \mathrm{the} \mathrm{same} \mathrm{degree}. \\ \mathbf{For}\mathbf{ }\mathbf{example}, 3{\mathrm{x}}^3+2{\mathrm{x}}^2\mathrm{y}+4{\mathrm{xy}}^2+{\mathrm{y}}^{3 }\mathrm{is} \mathrm{a} \mathrm{homogeneous} \mathrm{polynomial} \mathrm{of} \mathrm{degree} 3 \mathrm{in} \mathrm{two} \mathrm{variables}. \\ \mathrm{Here} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{exponents} \mathrm{in} \mathrm{each} \mathrm{term} \mathrm{is} \mathrm{always} 3. \\ \mathrm{The} \mathrm{other} \mathrm{examples} \mathrm{of} \mathrm{homogeneous} \mathrm{polynomials} \mathrm{are} \mathrm{as} \mathrm{follows}: \\ \left(1\right) {\mathrm{x}}^4+{\mathrm{y}}^4 \\ \left(2\right) {\mathrm{a}}^2+2\mathrm{ab}+{\mathrm{b}}^2 \\ \left(3\right) {\mathrm{x}}^5+{\mathrm{y}}^5+{\mathrm{xy}}^4+{\mathrm{x}}^4\mathrm{y}+{\mathrm{x}}^2{\mathrm{y}}^3+{\mathrm{x}}^3{\mathrm{y}}^2 \\ \left(4\right) {\mathrm{z}}^3+2\mathrm{xyz} \\ \mathrm{The} \mathrm{following} \mathrm{polynomials} \mathrm{are} \mathrm{not} \mathrm{homogeneous}: \\ {\mathrm{x}}^2+2\mathrm{x}+1, {\mathrm{x}}^2-9 \end{gathered}$$

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