degree

Examples of degree in the following topics:

• Basics of Graphing Polynomial Functions

  • A typical graph of a polynomial function of degree 3 is the following:
  • A polynomial of degree 6.
  • Its highest-degree coefficient is positive.
  • A polynomial of degree 5.
  • Its highest-degree coefficient is positive.

• 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 $2$.
  • Conversely, if the multiplicities of the roots of a polynomial add to its degree, and if its degree is at least $1$ (i.e. it is not constant), then it follows that it has at least one zero.
  • Therefore, 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).

• Radians

  • Radians are another way of measuring angles, and the measure of an angle can be converted between degrees and radians.
  • Recall that dividing a circle into 360 parts creates the degree measurement.
  • As stated, one radian is equal to $\displaystyle{ \frac{180^{\circ}}{\pi} }$ degrees, or just under 57.3 degrees ($57.3^{\circ}$).
  • Thus, to convert from radians to degrees, we can multiply by $\displaystyle{ \frac{180^\circ}{\pi} }$:
  • $\displaystyle{ \text{angle in degrees} = \text{angle in radians} \cdot \frac{180^\circ}{\pi} }$

• 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.
  • Notice that the answer is a polynomial of degree 3; this is also the highest degree of a polynomial in the problem.

• What Are Polynomials?

  • The degree of a polynomial $Q(x)$ is the highest degree of one of its terms.
  • For example, the degree of $P(x)$ is $13$.
  • 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.
  • Here the degree in $x$ of $x^3y^5$is $3$, the degree in $y$ of $x^3y^5$is $5$ and its joint degree or degree is $8$.

• Other Equations in Quadratic Form

  • Many equations with no odd-degree terms can be reduced to quadratics and solved with the same methods as quadratics.
  • Higher degree polynomial equations can be very difficult to solve.
  • For example, if a quartic equation is biquadratic—that is, it includes no terms of an odd-degree— there is a quick way to find the zeroes of the quartic function by reducing it into a quadratic form.  
  • Consider a quadratic function with no odd-degree terms which has the form:
  • If we let an arbitrary variable $p$ equal $x^2$, this can be reduced to an equation of a lower degree:

• Multiplying Polynomials

  • 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

• Finding Polynomials with Given Zeros

  • 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.
  • If we already count multiplicity in this number, than the degree equals the number of roots.
  • For example, if we are given two zeros, then a polynomial of second degree needs to be constructed.

• Polynomial and Rational Functions as Models

  • where $n$ (the degree of the polynomial) is an integer greater than or equal to $0$, $x$ and $y$ are variables, and $0\not=a_n,a_{n-1},\ldots,a_2, a_1$ and $a_0$ are constants.
  • Here, n and m define the degrees of the numerator and denominator, respectively, and together, they define the degree of the polynomial.
  • Polynomial curves generated to fit points (black dots) of a sine function: The red line is a first degree polynomial; the green is a second degree; the orange is a third degree; and the blue is a fourth degree.

• Simplifying Radical Expressions

  • where $n$ is the degree of the root.
  • A root of degree 2 is called a square root and a root of degree 3, a cube root.
  • Roots of higher degrees are referred to using ordinal numbers, as in fourth root, twentieth root, etc.