zero

Examples of zero 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 $2$.

• 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

• 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

• Polynomial Inequalities

  • The best way to solve a polynomial inequality is to find its zeros.
  • The three terms reveal zeros at $x=-3$, $x=-1$, and $x=2$.
  • The next zero is at $x=-1$.
  • At $x=-1$, $(x+1)$ 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 $A$ and $B$ are both not equal to zero.
  • Recall that a zero is a point at which a function's value will be equal to zero ($y=0$), and is the $x$-intercept of the function.
  • Therefore, the zero of the equation occurs at $x = \frac{5}{1} = 5$.
  • The zero is the point $(5, 0)$.
  • Example: Find the zero of the equation $3(y - 2) = \frac{1}{4}x +3$

• 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 $x$-coordinate at which the function's value equals zero.
  • If ${\Delta}$ 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.

• Zeros 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 $0$.
  • The zero of a function, $f(x)$, refers to the value or values of $x$ that will result in the function equaling zero, $f(x)=0$.
  • Use the quadratic equation and factoring methods to find the zeros of a polynomial

• 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 $f(x)=0$ is the $x$-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).