zeros

(noun)

In a given function, the values of $x$ at which $y = 0$, also called roots.

Related Terms

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