degree of a polynomial

(noun)

The highest value of an exponent placed on a variable in any of the terms of a polynomial.

Related Terms

Examples of degree of a polynomial in the following topics:

  • Basics of Graphing Polynomial Functions

    • A typical graph of a polynomial function of degree 3 is the following:
    • The graph of a degree 2 polynomial $f(x) = a_0 + a_1x + a_2x^2$, where $a_2 \neq 0$ is a parabola.
    • The graph of a degree 3 polynomial $f(x) = a_0 + a_1x + a_2x^2 + a_3x^3$, where $a_3 \neq 0$, is a cubic curve.
    • A polynomial of degree 6.
    • A polynomial of degree 5.

  • Finding Polynomials with Given Zeroes

    • One type of problem is to generate a polynomial from given zeros.
    • This can be solved using the property that if $x_0$ is a zero of a polynomial, then $(x-x_0)$ is a divisor of this polynomial and vice versa.
    • Remember that the degree of a polynomial, the highest exponent, dictates the maximum number of roots it can have.
    • Thus, the degree of a polynomial with a given number of roots is equal to or greater than the number of roots that are given.
    • For example, if we are given two zeroes, then a polynomial of second degree needs to be constructed.

  • Adding and Subtracting Polynomials

    • Note that any two polynomials can be added or subtracted, regardless of the number of terms in each, or the degrees of the polynomials.
    • The resulting polynomial will have the same degree as the polynomial with the higher degree in the problem.
    • You may be asked to add or subtract polynomials that have terms of different degrees.
    • See the second example below for a demonstration of this concept.
    • Notice that the answer is a polynomial of degree 3; this is also the highest degree of a polynomial in the problem.

  • The Fundamental Theorem of Algebra

    • Every polynomial of odd degree with real coefficients has a real zero.
    • So since the property is true for all polynomials of degree $0$, it is also true for all polynomials of degree $1$.
    • And since it is true for all polynomials of degree $1$, it is also true for all polynomials of degree $2$.
    • The multiplicities of the complex roots of a nonzero polynomial with complex coefficients add to the degree of said polynomial.
    • This last remark, together with the alternative statement of the fundamental theorem of algebra, tells us that the parity of the real roots (counted with multiplicity) of a polynomial with real coefficients, must be the same as the parity of the degree of said polynomial, i.e. a polynomial of even degree admits an even number of real roots and a polynomial of odd degree admits an odd number of real roots (counted with multiplicity).

  • What Are Polynomials?

    • A monomial over $\mathbb{R}$ in a single variable $x$ consists of a non-negative power of $x$, multiplied with a nonzero constant $c \in \mathbb{R}.$ So a polynomial looks like
    • A polynomial over $\mathbb{R}$ is a finite sum of monomials over $\mathbb{R}$.
    • The degree of a polynomial $Q(x)$ is the highest degree of one of its terms.
    • The degree of the zero polynomial is defined to be $-\infty$.
    • The degree of a polynomial is defined in the same way as in the real case.

  • Introduction to Factoring Polynomials

    • A polynomial consists of a sum of monomials.
    • However, sometimes it will be more useful to write a polynomial as a product of other polynomials with smaller degree, for example to study its zeros.
    • The process of rewriting a polynomial as a product is called factoring.
    • In all cases, a product of simpler objects than the original (smaller integers, or polynomials of smaller degree) is obtained.
    • is a factorization of a polynomial of degree $3$ into $3$ polynomials of degree $1$.

  • Finding Zeroes of Factored Polynomials

    • The factored form of a polynomial can reveal where the function crosses the $x$-axis.
    • Regardless of odd or even, any polynomial of positive order can have a maximum number of zeros equal to its order.
    • In general, we know from the remainder theorem that $a$ is a zero of $f(x)$ if and only if $x-a$ divides $f(x).$ Thus if we can factor $f(x)$ in polynomials of as small a degree as possible, we know its zeros by looking at all linear terms in the factorization.
    • Graph of the cubic function $f(x) = x^3 + 2x^2 - 5x - 6 = (x+3)(x+1)(x-2).$ We see that its roots equal the negative second coefficients of its first degree factors.
    • Use the factored form of a polynomial to find its zeros

  • Dividing Polynomials

    • Polynomial long division is a method for dividing a polynomial by another polynomial of the same or lower degree.
    • Subtract the product just obtained from the appropriate terms of the original dividend (remember that subtracting something having a minus sign is equivalent to adding something having a plus sign): $(x^3 − 12x^2) − (x^3 − 3x^2) = −12x^2 + 3x^2 = −9x^2$.
    • Polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree.This method is a generalized version of the familiar arithmetic technique called long division.It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.
    • Subtract the product just obtained from the appropriate terms of the original dividend (being careful that subtracting something having a minus sign is equivalent to adding something having a plus sign): $(x^3 − 12x^2) − (x^3 − 3x^2) = −12x^2 + 3x^2 = −9x^2$.
    • The calculated polynomial is the quotient, and the number left over (−123) is the remainder:

  • Multiplying Polynomials

    • Multiplying a polynomial by a monomial is a direct application of the distributive and associative properties.
    • So for the multiplication of a monomial with a polynomial we get the following procedure:
    • To multiply a polynomial $P(x) = M_1(x) + M_2(x) + \ldots + M_n(x)$ with a polynomial $Q(x) = N_1(x) + N_2(x) + \ldots + N_k(x)$, where both are written as a sum of monomials of distinct degrees, we get
    • Notice that since the highest degree term of $P(x)$ is multiplied with the highest degree term of $Q(x)$ we have that the degree of the product equals the sum of the degrees, since
    • So the roots of a product of polynomials are exactly the roots of its factors, i.e.

  • Polynomial Inequalities

    • Like any other function, a polynomial may be written as an inequality, giving a large range of solutions.
    • The easiest way to find the zeros of a polynomial is to express it in factored form.
    • The product of a positive and two negatives is positive, so we can conclude that the polynomial becomes positive as it passes $x=-3$.
    • Graph of the third-degree polynomial with the equation $y=x^3+2x^2-5x-6$.
    • Solve for the zeros of a polynomial inequality to find its solution