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Mathematics LibreTexts

5.2: Introduction to Polynomials

  • Page ID
    18355
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    Learning Objectives

    • Identify a polynomial and determine its degree.
    • Evaluate a polynomial for given values of the variables.
    • Evaluate a polynomial using function notation.

    Definitions

    A polynomial is a special algebraic expression with terms that consist of real number coefficients and variable factors with whole number exponents.

    \(\color{Cerulean}{Examples\:of\:polynomials:}\)

    \(3x^{2}\quad 7xy+5\quad \frac{3}{2}x^{3}+3x^{2}-\frac{1}{2}x+1\quad 6x^{2}y-4xy^{3}-4xy^{3}+7\)

    Polynomials do not have variables in the denominator of any term.

    \(\color{Cerulean}{Examples\:that\:are\:not\:polynomials:}\)

    \(\frac{2x^{2}}{y} \quad 5\sqrt{x}+5\quad 5x^{2}+3x^{-2}+7\quad \frac{2}{x}-\frac{5}{y}=3\)

    The degree of a term in a polynomial is defined to be the exponent of the variable, or if there is more than one variable in the term, the degree is the sum of their exponents. Recall that \(x^{0}=1\); any constant term can be written as a product of \(x^{0}\) and itself. Hence the degree of a constant term is \(0\).

    Term Degree
    \(3x^{2}\) \(2\)
    \(6x^{2}y\) \(2+1=3\)
    \(7a^{2}b^{3}\) \(2+3=5\)
    \(8\) \(0\), since \(8=8x^{0}\)
    \(2x\) \(1\), since \(x=x^{1}\)

    Table 5.2.1

    The degree of a polynomial is the largest degree of all of its terms.

    Polynomial Degree
    \(4x^{5}-3x^{3}+2x-1\) \(5\)
    \(6x^{2}y-5xy^{3}+7\) \(4\), because \(5xy^{3}\) has degree \(4\).
    \(12x+54\) \(1\), because \(x=x^{1}\)

    Table 5.2.2

    We classify polynomials by the number of terms and the degree as follows:

    Expression Classification Degree
    \(5x^{7}\) Monomial (one term) \(7\)
    \(8x^{6}-1\) Binomial (two terms) \(6\)
    \(-3x^{2}+x-1\) Trinomial (three terms) \(2\)
    \(5x^{3}-2x^{2}+3x-6\) Polynomial (many terms) \(3\)

    Table 5.2.3

    In this text, we will call polynomials with four or more terms simply polynomials.

    Example \(\PageIndex{1}\)

    Classify and state the degree:

    \(7x^{2}−4x^{5}−1\).

    Solution:

    Here there are three terms. The highest variable exponent is \(5\). Therefore, this is a trinomial of degree \(5\).

    Answer:

    Trinomial; degree \(5\)

    Example \(\PageIndex{2}\)

    Classify and state the degree:

    \(12a^{5}bc^{3}\).

    Solution:

    Since the expression consists of only multiplication, it is one term, a monomial. The variable part can be written as \(a^{5}b^{1}c^{3}; hence its degree is \(5+1+3=9\).

    Answer:

    Monomial; degree \(9\)

    Example \(\PageIndex{3}\)

    Classify and state the degree:

    \(4x^{2}y−6xy^{4}+5x^{3}y^{3}+4\).

    Solution:

    The term \(4x^{2}y\) has degree \(3\); \(−6xy^{4}\) has degree \(5; 5x^{3}y^{3}\) has degree \(6\); and the constant term \(4\) has degree \(0\). Therefore, the polynomial has \(4\) terms with degree \(6\).

    Answer:

    Polynomial; degree \(6\)