zeros

Examples of zeros in the following topics:

• Zeroes of Linear Functions

  • A zero, or $x$-intercept, is the point at which a linear function's value will equal zero.
  • Zeros can be observed graphically.  
  • Because the $x$-intercept (zero) is a point at which the function crosses the $x$-axis, it will have the value $(x,0)$, where $x$ is the zero.
  • The zero is $(-4,0)$.  
  • The blue line, $y=\frac{1}{2}x+2$, has a zero at $(-4,0)$; the red line, $y=-x+5$, has a zero at $(5,0)$.  

• Finding Polynomials with Given Zeros

  • To construct a polynomial from given zeros, set $x$ equal to each zero, move everything to one side, then multiply each resulting equation.
  • One type of problem is to generate a polynomial from given zeros.
  • If it is not specified what the multiplicity of the zeros are, we want the zeros to have multiplicity one.
  • There are no other zeros, i.e. if a number is not mentioned in the problem statement, it cannot be a zero of the polynomial we find.
  • Two polynomials with the same zeros: Both $f(x)$ and $g(x)$ have zeros $0, 1$ and $2$.

• Absolute Zero

  • Absolute zero is universal in the sense that all matteris in ground state at this temperature .
  • To be precise, a system at absolute zero still possesses quantum mechanical zero-point energy, the energy of its ground state.
  • The zero point of a thermodynamic temperature scale, such as the Kelvin scale, is set at absolute zero.
  • Note that all of the graphs extrapolate to zero pressure at the same temperature
  • Explain why absolute zero is a natural choice as the null point for a temperature unit system

• Finding Zeros of Factored Polynomials

  • An $x$ -value at which this occurs is called a "zero" or "root. "
  • A polynomial function may have many, one, or no zeros.
  • All polynomial functions of positive, odd order have at least one zero (this follows from the fundamental theorem of algebra), while polynomial functions of positive, even order may not have a zero (for example $x^4+1$ has no real zero, although it does have complex ones).
  • Replacing $x$ with a value that will make either $(x+3),(x+1)$  or $(x-2)$ zero will result in $f(x)$ being equal to zero.
  • Use the factored form of a polynomial to find its zeros

• The Third Law

  • According to the third law of thermodynamics, the entropy of a perfect crystal at absolute zero is exactly equal to zero.
  • The third law of thermodynamics is sometimes stated as follows: The entropy of a perfect crystal at absolute zero is exactly equal to zero.
  • Entropy is related to the number of possible microstates, and with only one microstate available at zero kelvin the entropy is exactly zero.
  • Nernst proposed that the entropy of a system at absolute zero would be a well-defined constant.
  • In simple terms, the third law states that the entropy of a perfect crystal approaches zero as the absolute temperature approaches zero.

• The Third Law of Thermodynamics and Absolute Energy

  • The entropy of a system at absolute zero is typically zero, and in all cases is determined only by the number of different ground states it has.
  • Specifically, the entropy of a pure crystalline substance at absolute zero temperature is zero.
  • At zero temperature the system must be in a state with the minimum thermal energy.
  • At absolute zero there is only 1 microstate possible (Ω=1) and ln(1) = 0.
  • For the entropy at absolute zero to be zero, the magnetic moments of a perfectly ordered crystal must themselves be perfectly ordered.

• Zero-Coupon Bonds

  • Examples of zero-coupon bonds include U.S.
  • Treasury bills, U.S. savings bonds, and long-term zero-coupon bonds.
  • This creates a supply of new zero coupon bonds.
  • Zero coupon bonds may be long- or short-term investments.
  • Long-term zero coupon maturity dates typically start at 10 to 15 years.

• First Condition

  • The first condition of equilibrium is that the net force in all directions must be zero.
  • This means that both the net force and the net torque on the object must be zero.
  • Here we will discuss the first condition, that of zero net force.
  • For example, the net external forces along the typical x- and y-axes are zero.
  • The forces acting on him add up to zero.

• Zero-Order Reactions

  • The rate law for a zero-order reaction is rate = k, where k is the rate constant.
  • This is the integrated rate law for a zero-order reaction.
  • For a zero-order reaction, the half-life is given by:
  • [A]0 represents the initial concentration and k is the zero-order rate constant.
  • Use graphs of zero-order rate equations to obtain the rate constant and the initial concentration data

• Rational Inequalities

  • The zeros in the denominator are $x$-values are at which the rational inequality is undefined, the result of dividing by zero.
  • The numerator has zeros at $x=-3$ and $x=1$.
  • The denominator has zeros at $x=-2$ and $x=2$.
  • For $x$ values that are zeros for the numerator polynomial, the rational function overall is equal to zero.
  • Solve for the zeros of a rational inequality to find its solution