zeros

(noun)

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

Related Terms

  • axis of symmetry
  • vertex
  • parabola

Examples of zeros in the following topics:

  • Zeroes of Linear Functions

    • A zero, or xxx-intercept, is the point at which a linear function's value will equal zero.
    • Zeros can be observed graphically.  
    • Because the xxx-intercept (zero) is a point at which the function crosses the xxx-axis, it will have the value (x,0)(x,0)(x,0), where xxx is the zero.
    • The zero is (−4,0)(-4,0)(−4,0).  
    • The blue line, y=12x+2y=\frac{1}{2}x+2y=​2​​1​​x+2, has a zero at (−4,0)(-4,0)(−4,0); the red line, y=−x+5y=-x+5y=−x+5, has a zero at (5,0)(5,0)(5,0).  
  • Finding Polynomials with Given Zeroes

    • To construct a polynomial from given zeroes, set x equal to each zero, move everything to one side, then multiply each resulting equation.
    • If it is not specified what the multiplicity of the zeros are, we want the zeros to have multiplicity one.
    • one for every non-zero number ccc.
    • Given zeros 0,10, 10,1 and 222.
    • Two polynomials with the same zeros: Both f(x)f(x)f(x) and g(x)g(x)g(x) have zeros 0,10, 10,1 and 222.
  • Finding Zeroes of Factored Polynomials

    • An xxx -value at which this occurs is called a "zero" or "root. "
    • A polynomial function may have zero, one, or many zeros.
    • Replacing xxx with a value that will make either (x+3),(x+1)(x+3),(x+1)(x+3),(x+1)  or (x−2)(x-2)(x−2) zero will result in f(x)f(x)f(x) being equal to zero.
    • Thus if you have found such a factorization of a given function, you can be completely sure what zeros the zeros of that function are.
    • Use the factored form of a polynomial to find its zeros
  • Rational Inequalities

    • The zeros in the denominator are xxx values are at which the rational inequality is undefined, the result of dividing by zero.
    • The numerator has zeros at x=−3x=-3x=−3 and x=1x=1x=1.
    • The denominator has zeros at x=−2x=-2x=−2 and x=2x=2x=2.
    • For xxx 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
  • Polynomial Inequalities

    • The best way to solve a polynomial inequality is to find its zeros.
    • The three terms reveal zeros at x=−3x=-3x=−3, x=−1x=-1x=−1, and x=2x=2x=2.
    • The next zero is at x=−1x=-1x=−1.
    • At x=−1x=-1x=−1, (x+1)(x+1)(x+1) equals zero, becoming positive to the right.
    • Solve for the zeros of a polynomial inequality to find its solution
  • Linear Equations in Standard Form

    • where AAA and BBB are not both equal to zero.
    • Recall that a zero is a point at which a function's value will be equal to zero (y=0y=0y=0), and is the xxx-intercept of the function.
    • Therefore, the zero of the equation occurs at x=51=5x = \frac{5}{1} = 5x=​1​​5​​=5.
    • The zero is the point (5,0)(5, 0)(5,0).
    • Find the zero of the equation 3(y−2)=14x+33(y - 2) = \frac{1}{4}x +33(y−2)=​4​​1​​x+3.
  • Zeroes of Polynomial Functions with Real Coefficients

    • A root, or zero, of a polynomial function is a value that can be plugged into the function and yield zero.
    • The zero of a function, f(x)f(x)f(x), refers to the value or values of xxx that will result in the function equaling zero, f(x)=0f(x)=0f(x)=0.
    • Use the quadratic equation and factoring methods to find the zeros of a polynomial
  • Zeroes of Polynomial Functions With Rational Coefficients

    • In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
    • Zero divided by any other integer equals zero.
    • Therefore zero is a rational number, but division by zero is undefined.
    • By setting each term to zero, it can be found that the zeros for this equation are x=-6 and x=-9/2.
    • Extend the techniques of finding zeros to polynomials with rational coefficients
  • The Discriminant

    • That is, it is the xxx-coordinate at which the function's value equals zero.
    • If Δ{\Delta}Δ is equal to zero, the square root in the quadratic formula is zero:
    • Since adding zero and subtracting zero in the quadratic equation lead to the same outcome, there is only one distinct root of the quadratic function.
    • Because Δ is greater than zero, the function has two distinct, real roots.
    • Because the value is greater than 0, the function has two distinct, real zeros.
  • Basics of Graphing Polynomial Functions

    • Between two zeros (and before the smallest zero, and after the greatest zero) a function will always be either positive, or negative.
    • Every time we cross a zero of odd multiplicity (if the number of zeros equals the degree of the polynomial, all zeros have multiplicity one and thus odd multiplicity) we change sign.
    • The graph of the zero polynomial f(x)=0f(x)=0f(x)=0 is the xxx-axis, since all real numbers are zeros.
    • It has exactly 6 zeroes and 5 local extrema.
    • It has 3 real zeros (and two complex ones).
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