Math Lesson 11.1.8 - The Homogenous and Non-Homogenous Polynomials

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Welcome to our Math lesson on The Homogenous and Non-Homogenous Polynomials, this is the eighth lesson of our suite of math lessons covering the topic of The Definition of Monomials and Polynomials, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Homogenous and Non-Homogenous Polynomials

When a multivariable polynomial has the same degree in all its terms, it is called homogenous. For example,

is a homogenous polynomial because all its terms are of the fourth degree.

On the other hand, the polynomial

is not homogenous, as its last term has not the same degree as the rest of the terms (it is a zero-degree monomial, while all the rest of the terms are fourth-degree monomials). Hence, knowing the degree of a polynomial has another advantage, as this helps in finding whether the given polynomial expression is homogeneous or not.

Example 8

Which of the following polynomials is homogenous?

1. P(x,y,z) = 5x - 2y - 18z
2. P(a,b,c) = 4a⁵ bc³ - 2/7 ab⁶ c² - c⁹
3. P(m,n) = 2m³ - m² n - mn² - 4mn

Solution 8

1. P(x, y, z) = 5x - 12y -18z is a homogenous polynomial, as all its component terms, 5x, 2y and 18z, are of the first degree, despite having different variables.
2. P(a,b,c) = 4a⁵ bc³ - 2/7 ab⁶ c² - c⁹ is a homogenous polynomial too, as all its component terms, 4a⁵bc³, -(2/7)ab6c2 and -c9, are of the ninth degree (5 + 1 + 3 = 1 + 6 + 2 = 9), despite the fact that the power of each individual variable is different.
3. P(m, n) = 2m³ - m²n - mn² - 4mn is not a homogenous polynomial as not all its component terms have the same degree. Thus, the first term (2m³) is of the third degree; the second term (-m²n) is also of the third degree (2 + 1 = 3); yet, the third term (-mn²) is still of the third degree (1 + 2 = 3). However, the fourth term (-4mn) is of the second degree (1 + 1 = 2). Hence, the given polynomial is not homogenous.

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