Ninth Grade Mathematics: Classification by Degree and Terms
Polynomials can be categorized and understood through two complementary perspectives: by counting the number of terms they contain, and by determining the highest power of the variable present. In this chapter, you will master the classification of polynomials by degree and terms, understand multivariable degrees, and explore the unique properties of the zero polynomial.
Determining the Degree of a Polynomial
The degree of a polynomial in a single variable is the highest exponent of the variable appearing in any non-zero term of the polynomial. The term containing this highest power is the leading term, and its coefficient is the leading coefficient.
Identifying the Degree:
P(x) = 7*x^4 - 2*x^6 + 11*x^2 - 5
Exponents present: 4, 6, 2, 0.
Highest exponent: 6.
Degree: 6
Leading Term: -2*x^6 (Leading Coefficient: -2)
In a polynomial containing multiple variables within each term, the degree of a term is the sum of the exponents of all the variables in that specific term. The degree of the polynomial is the maximum term degree found across the entire expression.
Multivariable Polynomial:
P(x, y) = 4*x^2*y^3 - 7*x^4*y^2 + 9*x*y - 12
Term 1: 4*x^2*y^3 ---> Exponent sum: 2 + 3 = 5
Term 2: -7*x^4*y^2 ---> Exponent sum: 4 + 2 = 6 <--- Highest degree
Term 3: 9*x*y ---> Exponent sum: 1 + 1 = 2
Term 4: -12 ---> Exponent sum: 0
Degree of P(x, y) = 6
Classification by Degree
Polynomials are classified into standard names based on their degree:
Degree Name of Polynomial Standard General Form Example
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0 Constant Polynomial P(x) = c (c != 0) P(x) = 7
1 Linear Polynomial P(x) = a*x + b P(x) = 3*x - 5
2 Quadratic Polynomial P(x) = a*x^2 + b*x + c P(x) = 2*x^2 + 4*x - 1
3 Cubic Polynomial P(x) = a*x^3 + b*x^2 + c*x + d P(x) = x^3 - 8
4 Quartic (Biquadratic) P(x) = a*x^4 + ... P(x) = 5*x^4 - 3*x + 2
5 Quintic Polynomial P(x) = a*x^5 + ... P(x) = x^5 + 4
The Zero Polynomial
The number 0 can be viewed as a constant polynomial, P(x) = 0. However, unlike non-zero constant polynomials which have degree 0, the degree of the zero polynomial is officially undefined (or defined as negative infinity in advanced algebra). This is because 0 can be written as 0x^1, 0x^2, 0x^10, or 0x^100 without changing its value, making it impossible to assign a single unique maximum exponent.
Classification by the Number of Terms
Independent of its degree, a polynomial can also be classified by how many non-zero terms it contains when fully simplified and combined:
Number of Terms Name Example Analysis
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1 Monomial 7*x^4 Single term
2 Binomial 3*x^2 - 8 Two distinct terms
3 Trinomial x^2 - 5*x + 6 Three distinct terms
4 or more Polynomial 2*x^3 - 4*x^2 + 3*x - 7 Four or more terms
Before classifying by the number of terms, you must always combine like terms. For example, 4x^2 + 3x - 2x^2 + 5 appears to have four terms, but simplifies to 2x^2 + 3*x + 5, which is a trinomial.
Chapter Exercises
Exercise 1: Classify each of the following polynomials both by degree and by number of terms after simplifying: P(x) = 5x^3 - 2x + 4, Q(y) = 7y^2 - 3 + 4y^2 + 9, R(z) = 14z^4 - 2z^4, and S(t) = 8*t - 15.
Exercise 2: Determine the degree of the multivariable polynomial P(a, b, c) = 3a^2b^3c - 5a^4b^2 + 8abc^4 - 11.
Exercise 3: Explain the essential distinction between the degree of a non-zero constant polynomial such as P(x) = -12 and the degree of the zero polynomial P(x) = 0.
Solutions and Explanations
Solution 1: For P(x) = 5x^3 - 2x + 4, the highest power of x is 3, making it a cubic polynomial; it contains three non-like terms, making it a trinomial. For Q(y) = 7y^2 - 3 + 4y^2 + 9, combining like terms yields (7 + 4)y^2 + (-3 + 9) = 11y^2 + 6; the highest degree is 2 (quadratic) and it contains two terms, making it a quadratic binomial. For R(z) = 14z^4 - 2z^4, combining like terms gives 12z^4; this has degree 4 (quartic) and consists of a single term, making it a quartic monomial. For S(t) = 8t - 15, the highest power of t is 1 (linear) and there are two terms, making it a linear binomial.
Solution 2: To find the degree of P(a, b, c) = 3a^2b^3c - 5a^4b^2 + 8abc^4 - 11, compute the degree of each individual term by summing its variable exponents. For the first term 3a^2b^3c^1, the sum is 2 + 3 + 1 = 6. For the second term -5a^4b^2, the sum is 4 + 2 = 6. For the third term 8a^1b^1c^4, the sum is 1 + 1 + 4 = 6. The constant term -11 has degree 0. The highest term degree found in the polynomial is 6; therefore, the degree of P(a, b, c) is 6.
Solution 3: A non-zero constant polynomial, such as P(x) = -12, can be written uniquely in terms of the variable as -12x^0. Since x^0 = 1 for any non-zero x, the highest exponent associated with a non-zero coefficient is unambiguously 0. By contrast, for the zero polynomial P(x) = 0, any power of x can be attached with a coefficient of zero: 0 = 0x^0 = 0x^1 = 0x^5 = 0*x^n for every integer n. Because no unique non-zero leading coefficient exists to anchor the maximum exponent, the degree of the zero polynomial cannot be determined and is formally defined as undefined.